Mathematics High School

## Answers

**Answer 1**

a) The **rate of change** of the number of downloads is given by the **derivative **of the function f(t).

[tex]f(t) = 3,000,000/(1 + 400e^{(-0.55t)})[/tex] Differentiating both sides of the function with respect to t, we get;

[tex]f'(t) = 3,000,000 × [d/dt (1 + 400e^{(-0.55t)})] / (1 + 400e^{(-0.55t)})^2[/tex]

Using the** chain rule**, we get;

[tex]d/dt (1 + 400e^{(-0.55t)}) = 0 + 400 × (-0.55) × e^{(-0.55t)} = -220e^{(-0.55t)[/tex]

Therefore;

[tex]f'(t) = -1650000000e^{(-0.55t)} / (1 + 400e^{(-0.55t)})^2[/tex]

To find the time when the rate of change of downloads is **maximized**, we need to find where the derivative equals 0. Therefore;

[tex]f'(t) = 0 = -1650000000e^{(-0.55t)}/ (1 + 400e^{(-0.55t)})^2[/tex]

Multiplying both sides by [tex](1 + 400e^{(-0.55t)})^2[/tex], we get;

[tex]0 = -1650000000e^{(-0.55t)[/tex]

Therefore;

[tex]e^{(-0.55t) }[/tex]= 0t = ∞

Therefore, there is no maximum rate of change of downloads.

b) The rate of change of downloads is given by the derivative of the **function **f(t).

Therefore, using the derivative found in part a;

[tex]f'(t) = -1650000000e^{(-0.55t)} / (1 + 400e^{(-0.55t)})^2[/tex]

Substituting t = ∞, we get;

[tex]f'(∞) = -1650000000e^{(-0.55 × ∞)} / (1 + 400e^{(-0.55 × ∞)})^2 = 0c)[/tex]

To find the number of times the song has been downloaded at time t, we need to evaluate the original function f(t) given by;

[tex]f(t) = 3,000,000/(1+400e^{(-0.55t)})[/tex]

Substituting t = ∞, we get;

[tex]f(∞) = 3,000,000 / (1 + 400e^{(-0.55 × ∞)}) = 3,000,000[/tex]

Therefore, the song has been downloaded 3,000,000 times at time t = ∞.

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**Answer 2**

The song has been **downloaded **approximately 7481 times at t = 0. The given **function **is:

f(t) = 3,000,000/(1+400[tex]e^{(-0.55t[/tex]))

a) To find the maximum rate of change of the number of downloads, we need to find the **derivative **of f(t).

f(t) = 3,000,000/(1+400[tex]e^{(-0.55t)[/tex])

Differentiating w.r.t t, we get:

f'(t) = [d/dt] [3,000,000/(1+400[tex]e^{(-0.55t)[/tex])]

f'(t) = 3,000,000 [d/dt] [1/(1+400[tex]e^{(-0.55t)[/tex])]

f'(t) = 3,000,000 [400[tex]e^{(-0.55t)[/tex] * 0.55] / (1+400[tex]e^{(-0.55t)[/tex])²

The rate of change is **maximum **when the first derivative is equal to 0.So, equating the first derivative to 0, we get:

3,000,000 [400[tex]e^{(-0.55t)[/tex] * 0.55] / (1+400[tex]e^{(-0.55t)})^2[/tex] = 0

Therefore, 400[tex]e^{(-0.55t)[/tex] * 0.55 = 0[tex]e^{(-0.55t)[/tex] = 0t = 0

So, the maximum rate of change of the number of downloads is achieved at t = 0.

b) To find the rate of change of the number of downloads at t = 0, we need to evaluate f'(0).

f'(t) = 3,000,000 [400[tex]e^{(-0.55t)[/tex]* 0.55] / (1+400[tex]e^{(-0.55t)[/tex])²

f'(0) = 3,000,000 [400[tex]e^{(0)[/tex] * 0.55] / (1+400[tex]e^{(0)[/tex])²

f'(0) = 165,000

So, the rate of change of the number of downloads at t = 0 is 165,000.

c) To find the **number **of times the song has been downloaded at t = 0, we need to evaluate f(0).

f(t) = 3,000,000/(1+400[tex]e^{(-0.55t)[/tex])

f(0)= 3,000,000/(1+400[tex]e^{(0)[/tex])

f(0) = 3,000,000/401

f(0) = 7481.3

So, the song has been downloaded approximately 7481 times at t = 0.

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## Related Questions

A body moves on a coordinate line such that it has a position s=f(t)=12-4t+3 on the interval 05158, with s in meters and in seconds. a. Find the body's displacement and average velocity for the given time interval b. Find the body's speed and acceleration at the endpoints of the interval c. When, if ever, during the interval does the body change direction?

### Answers

a. **Displacement **= -59.4 meters, Average velocity = -4.29 m/s

b. Speed (start) = 4 m/s, Speed (end) = 4 m/s, **Acceleration **(start) = 0 m/s, Acceleration (end) = 0 m/s c.

The body never changes direction during the given time interval.

a. To find the body's **displacement**,

We need to find the change in **position **(s) over the given time **interval**. So, we have:

**displacement **= s(end) - s(start)

displacement = f(15.8) - f(0.5)

displacement = (12 - 4(15.8) + 3) - (12 - 4(0.5) + 3)

displacement = -59.4 meters

To find the **average **velocity, we need to find the total distance traveled **divided **by the time **interval**.

So, we have:

average velocity = displacement / time interval **average **velocity

= -59.4 / (15.8 - 0.5)

average **velocity **= -4.29 m/s

b. To find the body's speed and **acceleration **at the endpoints of the interval, we need to **differentiate **the given position **function **(f(t)) with respect to time (t).

So, we have:

**speed **= |f'(t)|

speed(start) = |f'(0.5)|

= |-4|

= 4 m/s

speed(end) = |f'(15.8)|

= |-4|

= 4 m/s

Acceleration = f''(t)

Acceleration(start) = f''(0.5) = 0 m/s²

Acceleration(end) = f''(15.8) = 0 m/s^2

c. To find when the body changes **direction**, we need to find where the velocity (f'(t)) changes **sign**.

So, we need to solve the **equation **f'(t) = 0. We have:

f'(t) = -4 -4 = 0 This equation has no **solutions**,

This means the **velocity **never changes sign and the body never changes direction.

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Use the definition of the derivative to find f'(x) where f(x) = √9- -

### Answers

We used the **definition **of the **derivative** to find f'(x) where f(x) = √(9 - x²). The derivative of f(x) is f′(x) = -2x / √(9 - x²).

The given function is f(x) = √(9 - x²)

Using the definition of derivative,f′(x) = lim_(h → 0) [f(x + h) - f(x)] / h

Let's begin by finding f(x + h) = √(9 - (x + h)²)f(x + h) = √[9 - x² - 2xh - h²]

Now, subtracting f(x) = √(9 - x²) from f(x + h) = √[9 - x² - 2xh - h²]f(x + h) - f(x) = √[9 - x² - 2xh - h²] - √(9 - x²)

Next, multiply the numerator and denominator by the conjugate of the numerator,

f(x + h) - f(x) = √[9 - x² - 2xh - h²] - √(9 - x²) * √[9 - x² - 2xh - h² + 2xh] / √[9 - x² - 2xh - h² + 2xh]f(x + h) - f(x) = (9 - x² - 2xh - h²) - (9 - x²) / [h * √[9 - x² - 2xh - h² + 2xh] + √(9 - x²)]f(x + h) - f(x) = -2xh - h² / [h * √[9 - x² - 2xh - h² + 2xh] + √(9 - x²)]

Simplifying,f′(x) = lim_(h → 0) -2xh - h² / [h * √[9 - x² - 2xh - h² + 2xh] + √(9 - x²)]

Dividing the **numerator **and the **denominator **by h,f′(x) = lim_(h → 0) -2x - h / [√[9 - x² - 2xh - h² + 2xh] / h + √(9 - x²) / h]

Finally, taking the limit of the above expression as h tends to 0,f′(x) = -2x / √(9 - x²)

The derivative of a function is the measure of its rate of change with respect to the change in its independent variable. The derivative is a fundamental concept in calculus and is used to find the slopes of curves and the tangent lines of functions. The definition of the derivative is given by,f′(x) = lim_(h → 0) [f(x + h) - f(x)] / h

This definition can be used to find the derivative of any function. The given function is f(x) = √(9 - x²). Applying the definition of the derivative,

f′(x) = lim_(h → 0) [f(x + h) - f(x)] / h

Let's begin by finding f(x + h) = √(9 - (x + h)²).

Subtracting f(x) = √(9 - x²) from f(x + h) = √[9 - x² - 2xh - h²], we get,f(x + h) - f(x) = √[9 - x² - 2xh - h²] - √(9 - x²)

Multiplying the numerator and denominator by the **conjugate **of the numerator,f(x + h) - f(x) = √[9 - x² - 2xh - h²] - √(9 - x²) * √[9 - x² - 2xh - h² + 2xh] / √[9 - x² - 2xh - h² + 2xh]

Simplifying the expression, we get,f′(x) = -2x / √(9 - x²)Hence, the derivative of f(x) = √(9 - x²) is f′(x) = -2x / √(9 - x²)

Therefore, we used the definition of the derivative to find f'(x) where f(x) = √(9 - x²). The derivative of f(x) is f′(x) = -2x / √(9 - x²).

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If S is a subset of a real vector space V , answer true or

false.

a) \( S \cap S^{\perp}=\{0\} \) \( S \subseteq\left(S^{\perp}\right)^{\perp} \) \( \left(S^{\perp}\right)^{\perp}=S \) 1) \( \left(S^{\perp}\right)^{\perp}=\operatorname{Span}(S) \) \( \left[\left(S^{

### Answers

a) S ∩ S⊥ = {0} is true.b) S ⊆ (S⊥)⊥ is true.c) (S⊥)⊥ ≠ S is false. The statements (a) and (b) are true, while the statement (c) is false. S is a subset of a **real vector** space V.

S is a subset of a real vector space V, and we need to answer whether the given statements are true or false.

Let's look at each of them one by one:a) S ∩ S⊥ = {0}. This statement is true because S and S⊥ have no non-zero vectors in common.

So, their **intersection **contains only the zero vector, {0}. Therefore, S ∩ S⊥ = {0}.b) S ⊆ (S⊥)⊥This statement is also true because the orthogonal complement of S, S⊥, contains all vectors that are orthogonal to S. So, if we take the orthogonal complement of S⊥, we get a set that contains all vectors that are orthogonal to S⊥, which means they are parallel to S. Therefore, (S⊥)⊥ contains all vectors that are **parallel** to S, which includes all vectors in S.

Hence, S is a subset of (S⊥)⊥.c) (S⊥)⊥ = SThis statement is false because the orthogonal complement of any subset of a vector space contains all vectors that are **orthogonal **to that subset, but it does not necessarily contain all vectors in the vector space. So, taking the orthogonal complement of (S⊥), which contains all vectors orthogonal to S, gives a set that contains all vectors that are parallel to S.

However, it does not contain all vectors in the vector space V, which means it does not contain all vectors in S. Therefore, (S⊥)⊥ is a **superset** of S but not equal to S.

Hence, (S⊥)⊥ ≠ S. In conclusion, we have: a) S ∩ S⊥ = {0} is true.b) S ⊆ (S⊥)⊥ is true.c) (S⊥)⊥ ≠ S is false.

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An iterative formula is shown below.

Starting with x₁

Xn+1 = Xn + 3

=

21, work out the value of 2.

### Answers

**Answer:**

the value of x_2 is,

[tex]x_{2} = 24[/tex]

**Step-by-step explanation:**

We have the formula,

[tex]x_{n+1} = x_{n} + 3[/tex]

And we are given that,

[tex]x_{1} = 21[/tex]

Then finding x_2, we have

[tex]x_{1+1} = x_{1} + 3\\x_{2} = x_{1} + 3\\\\since , x_{1} = 21, we \ have \\x_{2} = 21 + 3\\x_{2} = 24[/tex]

So,

[tex]x_{2} = 24[/tex]

Evaluate each of the following. 5! / 2! 2!

### Answers

the **value** of 5! / (2! * 2!) is 30.this is **final** answer.

To evaluate the **expression** 5! / (2! * 2!), we first calculate the factorials involved.

The factorial of a number is the product of that number and all positive integers below it. Let's calculate the **factorials**:

5! = 5 * 4 * 3 * 2 * 1 = 120

2! = 2 * 1 = 2

Now, let's substitute these values **back** into the expression:

5! / (2! * 2!) = 120 / (2 * 2)

Simplifying further:

120 / (2 * 2) = 120 / 4 = 30

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Assume x and y are functions of t. Evaluate dtdy for the following. y3=2x4+6;dtdx=4,x=1,y=2 dtdy= (Round to two decimal places as needed.)

### Answers

option (a) is correct.Using the **chain** rule of differentiation, we found dtdy for the given function y³ = 2x⁴ + 6. We **substituted** the given values of x and y and found that dtdy = 2.

Given the **equation** y³ = 2x⁴ + 6, we need to evaluate dtdy for the given values of x, y and dtdx.We are given dtdx = 4, x = 1 and y = 2. Now, we need to **evaluate** dtdy. We can use the chain rule of differentiation to find dtdy.Using the chain rule, we have; dtdy = dtdx * dxdyWe are given dtdx = 4. To find dxdy, we need to find dydx first.dydx can be found by differentiating the given equation with respect to x as follows: y³ = 2x⁴ + 6y² * dydx = 8x³dydx = 8x³/y²Now, we can find dxdy by inverting dydx as follows: dydx = 8x³/y² => dxdy = y²/8x³Substituting the given values of x and y in the above equation, we have; dtdy = 4 * y²/8x³ = 4 * 2²/8*1³= 4 * 1/2 = 2 Therefore, 2. Hence, option (a) is correct.

Using the chain **rule** of differentiation, we found dtdy for the given function y³ = 2x⁴ + 6. We substituted the given values of x and y and found that dtdy = 2.

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makes that are defined mathematics as geometrically definable shapes that can be easily scaled and rotated without losing their shape and identity are known as

### Answers

**Mathematical objects** that can be easily scaled and **rotated **without losing their **shape **and **identity **are known as "**rigid bodies.**"

In **geometry**, rigid bodies are defined as objects that **maintain **their **shape **and **size **under rigid motions, which include **translations **(shifting) and rotations.

These objects possess properties such as symmetry, **congruence**, and invariance under **transformations**. Examples of rigid bodies include circles, squares, **cubes**, and **spheres**.

The ability to scale and rotate rigid bodies while preserving their geometric properties makes them useful in various mathematical **applications**, such as **coordinate **transformations, geometric proofs, and **modeling physical **objects in **engineering **and **physics**.

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6. A school is having a canned food drive. Each class is challenged to collect 140 cans. If 8

classes meet the challenge, how many cans will the school collect?

Equation:

Solution:

### Answers

Equation: 140 x 8 = 1120

Solution = 1120 cans

please show steps. thanks

1. [5 Points] Simplify e3 In z - 24 - 4x + 2 X

### Answers

The **equation** e³ in z - 24 - 4x + 2x **simplifies** to e^(3i*z) - 24 - 2x.

We are to simplify the expression e³in z - 24 - 4x + 2x.

Recall that e is the base of natural **logarithms** and has a **constant** value of approximately 2.71828.

Simplifying e³in z - 24 - 4x + 2x.

First, we should note that e³in z means e raised to the **power** of 3i * z. We then have the expression as follows:

e^(3i*z) - 24 - 2x

Therefore, we conclude that e³in z - 24 - 4x + 2x simplifies to e^(3i*z) - 24 - 2x.

Thus the solution is:

e³in z - 24 - 4x + 2x = e^(3i*z) - 24 - 2x.

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2n 3n+ 1 15. Let an (a) Determine whether {a} is convergent. (b) Determine whether Σ-1a, is convergent.

### Answers

Given the series 2n 3n+ 1 15, we can **determine whether **the sequence {a} is convergent and also determine whether the series Σ-1a, is convergent .Part a) Determine whether {a} is convergent The nth term in the **sequence** is given by;`

an = 2n/(3n + 1) + 15`Dividing by n, we have; `[tex]an/n = 2/(3 + 1/n) + 15/n[/tex] `As n approaches infinity, 1/n approaches 0. Thus the second term tends to 0 and vanishes. Hence;`lim (an/n) = lim(2/(3 + 0)) = 2/3`Since the limit of the ratio of successive terms in the sequence exists and is not equal to 1, the sequence {a} is** convergent.** Part b) Determine whether Σ-1a, is convergent.

Rewriting the nth term in the series, we get;`1/an = 1/(2n/(3n + 1) + 15)`Multiplying and dividing the term by 3n + 1, we have;`[tex]1/an [tex]= 3n + 1/2n(3n + 1) + 15(3n + 1)[/tex]`Splitting the fraction into partial fractions, we get;`1/an = (3/2)[1/n - (n + 1)/(3n + 1)] - 1/10[/tex]`We can now use the **comparison test **to determine the convergence of the series Σ1/an .The series Σ1/n diverges. The series Σ(n + 1)/(3n + 1) converges. (We can prove this using the ratio test)Therefore, by the comparison test, the series Σ1/an also **diverges**.

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1 1 The formula 1= x+22 models the percentage of households with an interfaith marriage, 1, x years after 1989. The formula N=x+9 models t 1989 Answer parts a-d below. a. In which years will more than

### Answers

a. In 11 years, more than 33% of households will have an **interfaith marriage**. b. More than 16% of households will have a person of faith married to someone with no religion in 7 years after 1998,

a. In which years will more than 33% of households have an interfaith marriage?

After the year 2033, more than 33% of households will have an interfaith marriage. Substituting the value of the **formula **1=x+22;

where 1 is the 1% of households with interfaith marriage and 22 is the number of years after 1989, we get, 33=x+22

Therefore, x = 33 - 22 = 11

Therefore, more than 33% of households will have an interfaith marriage in 11 years after 2022, i.e., after the year 2033.

b. In which years will more than 16% of **households **have a person of faith married to someone with no religion?

After the year 1998, more than 16% of households will have a person of faith married to someone with no religion. Substituting the value of the formula N=x+9;

where N is the **percentage **of households in which a person of faith is married to someone with no religion and 9 is the number of years after 1989, we get, 16=x+9

Therefore, x = 16 - 9 = 7

Therefore, more than 16% of households will have a person of faith married to someone with no religion in 7 years after 1998, i.e., after the year 2005.

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This proportion can be solved to find the unknown length, x, in the larger triangle. 34=7.5x

### Answers

The **proportion** 34 = 7.5x is given, and it can be solved to find the unknown **length**, x, in the larger triangle.

To solve the proportion 34 = 7.5x, we can apply the principles of solving proportions. In proportion, the **cross-products** are **equal**. So, we can cross-multiply the given proportion as follows:

34 × 1 = 7.5x × 1

Simplifying the **equation**, we have:

34 = 7.5x

To isolate x, we divide both sides of the equation by 7.5:

34 / 7.5 = x

Evaluating the division, we find:

x ≈ 4.533

Therefore, the unknown length, x, in the larger **triangle** is approximately 4.533. This value satisfies the given proportion, and it represents the ratio between the lengths of the sides in the triangle.

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Problem 7. Use integration by parts to evaluate the definite integral f'te ¹ dt.

### Answers

We are evaluating a definite integral, we can use the limits of integration to find the exact value of the **integral**. However, the integral we have does not have any limits of integration given, so we cannot determine an exact value for it.

Integration by parts is a way of integrating a product of two functions. The method **involves** differentiating one function and integrating the other. The formula is: ∫f(x)g'(x)dx = f(x)g(x) - ∫g(x)f'(x)dx To evaluate the definite integral f'(t)e¹dt using integration by parts, we choose u = e¹ and

dv/dt = f'(t).

Then, we have du/dt = 0

and v = f(t).

Now, using the **formula** above, we have: f(t)e¹ - ∫0f(t)dt = f(t)e¹ + C, where C is the constant of integration. Since we are evaluating a definite integral, we can use the limits of integration to find the exact value of the integral. However, the integral we have does not have any limits of integration given, so we cannot determine an exact value for it. Therefore, the main answer to this problem is: f(t)e¹ + C, where C is the constant of integration.

Choose u and dv/dt The first step in integration by parts is to choose u and dv/dt. Here, we choose u = e¹ and

dv/dt = f'(t). Find du/dt and v To find du/dt, we **differentiate** u with respect to t.

Here, du/dt = 0, since e¹ is a constant.

To find v, we integrate dv/dt with respect to t.

Here, v = f(t). Therefore, we have: f(t)e¹ + C, where C is the constant of integration. Since we are evaluating a definite integral, we can use the **limits** of integration to find the exact value of the integral. However, the integral we have does not have any limits of integration given, so we cannot determine an exact value for it.

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In parallelogram ABCD, \overline{B D} and \overline{A C} intersect at E . If A E=9, B E=3 x-7 , and D E=x+5 , find x .

### Answers

The value of x in **parallelogram **ABCD is 16/3.

To find the value of x in **parallelogram **ABCD, we can use the property that **opposite sides **of a** parallelogram **are equal in length.

Given that AE = 9 and BE = 3x - 7, we can set up the equation AE = BE:

9 = 3x - 7

To isolate x, we can add 7 to both sides of the equation:

9 + 7 = 3x

16 = 3x

Divide both sides of the equation by 3:

16/3 = x

Therefore, x = 16/3.

In order to find the value of x, we need to know the exact value of x.

So, the value of x in **parallelogram **ABCD is 16/3.

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4. [12 pts.] Set up a linear system of equations for the following problem. Use any method to solve the system and remember to interpret your solution. The campus bookstore recently sold a total of 16

### Answers

The question is about **setting **up a linear system of equations for the **problem **given below. The campus bookstore recently sold a total of 164 textbooks and backpacks.

A backpack sells for $18, and a textbook sells for $76. The total sales **amounted **to $8,900. We are to use any method to solve the system and remember to interpret our solution.Set up a linear system of equationsTo set up a linear system of equations, we define the variables. Let x be the number of backpacks sold and y be the number of textbooks sold.

The first equation is given by the total number of backpacks and **textbooks **sold:x + y = 164The second equation is given by the total sales amount:18x + 76y = 8,900method:x + y = 164 ⟶ y = 164 - x18x + 76y = 8,900 ⟶ 18x + 76(164 - x) = 8,900⟹ 18x + 12,464 - 76x = 8,900⟹ -58x = -3,564⟹ x = 61.38 ⟶ 61y = 164 - x = 164 - 61 = 103Interpreting the solution.

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A baseball team plays in a stadium that holds 46000 spectators. With the ticket price at $12 the average attendance has been 19000. When the price dropped to $9, the average attendance rose to 23000. a) Find the demand function p(x), where x is the number of the spectators. (Assume p(x)is linear.) p(x) = b) How should ticket prices be set to maximize revenue? Price = $

### Answers

Price to maximize revenue: $10.25.

We know that the **demand function** is linear in nature and passes through two points i.e (19000, 12) and (23000, 9).

Now we can use the two-point form to find the equation of the line passing through these two points. The two-point form is: y - y1 = (y2 - y1)/(x2 - x1) * (x - x1)Here, x1 = 19000, y1 = 12, x2 = 23000, y2 = 9.

Plugging these values in the above equation, we get:

p(x) - 12 = (9 - 12)/(23000 - 19000) * (x - 19000)p(x) - 12

= -3/4000 * (x - 19000)p(x) - 12

= -3x/4000 + 57/2p(x)

= (-3/4000)x + 577/50

So, the demand function p(x) = (-3/4000)x + 577/50.

Revenue is given by: R = p(x) * x

So, we need to find the value of x at which the revenue is maximum.

For that, we can differentiate the revenue equation with respect to x and equate it to zero.

We get: dR/dx = p(x) + x * dp(x)/dx

= (-3/4000)x + 577/50 + x * (-3/4000)

= (-6/4000)x + 577

= 0

=> x = 19200/3 = 6400

We can find the price at which the revenue is maximum by substituting this value of x in the demand function.

We get:

p(x) = (-3/4000)x + 577/50

=> p(6400) = (-3/4000) * 6400 + 577/50= 10.25

Therefore, the price at which the revenue is maximum is $10.25 and the revenue is $65,536.

Price to maximize revenue: $10.25.

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Let W be the space of all 3×3 matrices of the form : ⎣⎡a000c0b0d⎦⎤, where a,b,c and d∈R. Does ⎣⎡10001010−1⎦⎤ belongs to W?

### Answers

Let W be the **space **of all 3×3 **matrices **of the form : ⎣⎡a000c0b0d⎦⎤, where a,b,c and d∈R. Does ⎣⎡10001010−1⎦⎤ belong to W?

Solution:The given matrix[tex]⎣⎡10001010−1⎦⎤[/tex]can be written as [tex]⎣⎡100⎦⎤ ⎣⎡010⎦⎤ ⎣⎡00−1⎦⎤[/tex]Let A= ⎣⎡a000c0b0d⎦⎤ belong to W where a, b, c, d ∈ R.

Let's check **whether **the given matrix A belongs to the set W or not.As per the definition of the set W, a matrix A is said to belong to W if and only if it can be written in the form ⎣⎡a000c0b0d⎦⎤.

Thus, **comparing **the given matrix A and the form of the matrix which belongs to W,we havea =[tex]1b = 0c = 0d = −1[/tex] Since a, b, c, d are all real **numbers**, hence the given matrix A belongs to the set W.

The given matrix [tex]⎣⎡10001010−1⎦⎤[/tex] **belongs **to the set W of all 3×3 matrices of the form [tex]⎣⎡a000c0b0d⎦⎤[/tex], where a, b, c, and d are real numbers.

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Find all excluded values for the expression. That is, find all values of \( x \) for which the expression is undefined. \[ \frac{x+6}{x^{2}-1} \] If there is more than one value, separate them with co

### Answers

Therefore, the excluded values for the expression are 1,-1.

The **expression** is given by:[tex]${\frac{x+6}{x^2-1} $[/tex] .The **denominator** can not be zero as division by zero is undefined. So, we can set it equal to zero as follows:[tex]$$\begin{aligned} x^2-1&=0\\ x^2&=1\\ x&=\pm1 \end{aligned}$$[/tex]

It's important to note that the concept of undefined in mathematics often arises from restrictions within a particular mathematical system or context. Mathematicians define functions and operations based on the properties they want to preserve and the goals of their mathematical framework.

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find the smallest integer a such that the intermediate value theorem guarantees that f(x) has a zero on the interval (−2,a). f(x)=−x2- 2x- 1

### Answers

Smallest **integer** a = -1.

We need to find the smallest integer a such that the intermediate value theorem guarantees that f(x) has a zero on the interval (−2,a).

f(x) = -x² - 2x - 1

We need to find the value of 'a'.

To find that we need to apply the** intermediate value theorem** .To apply intermediate value theorem, we need to check if the function changes sign in the given interval, i.e., if f(-2) and f(a) are of opposite signs. If they are of opposite signs, the **function** changes sign and there exists at least one **root** in the interval. We have:

f(x) = -x² - 2x - 1f(-2) = -(-2)² - 2(-2) - 1 = -3f(a) = -a² - 2a - 1

Now we need to find the value of 'a' such that f(-2) and f(a) are of **opposite signs**.

Substituting a = -1 in f(a), we have:

f(-1) = -(-1)² - 2(-1) - 1 = -2

This means f(-2) and f(-1) are of opposite signs. Hence, there exists a value of 'a' in (-2, -1) such that f(x) has a zero. Therefore, the smallest integer a such that the intermediate value theorem guarantees that f(x) has a zero on the interval (−2,a) is -1.

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The cost \( \mathrm{C} \) in dollars of manufacturing \( \mathrm{x} \) bicycles at a production plant is given by the function shown below. \[ C(x)=2 x^{2}-1000 x+138,500 \] a. Find the number of bicy

### Answers

To minimize the **cost**, the number of bicycles that must be manufactured is 250.

To find the **number **of bicycles that must be manufactured to minimize the cost, we can use calculus. The cost function C(x) is given by:

C(x) = 2x² - 1000x + 138,500

To minimize the cost, we need to find the critical points of the function, which occur where the derivative of the function is equal to zero.

Taking the **derivative **of C(x) with respect to x:

C'(x) = 4x - 1000

Setting C'(x) = 0 and solving for x:

4x - 1000 = 0

4x = 1000

x = 1000/4

x = 250

So, the **critical **point occurs at x = 250.

To determine whether this critical point is a minimum or maximum, we can examine the second derivative of C(x).

Taking the second derivative of C(x) with respect to x:

C''(x) = 4

Since the second derivative is **positive **(C''(x) > 0), we conclude that the critical point x = 250 corresponds to a minimum.

Correct Question:

The cost C in dollars of manufacturing x bicycles at a production plant is given by the function shown below.

C(x)=2x²-1000 x+138,500

Find the number of bicycles that must be manufactured to minimize the cost.

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solve the system of first-order linear differential equations. (use c1 and c2 as constants.) y1' = y1 3y2 y2' = 3y1 y2

### Answers

The solution to the system of first-order **linear** differential equations is given by y1 = 3c1e^(4t) - c2e^(-2t) and y2 = -c1e^(4t) + 3c2e^(-2t), where c1 and c2 are constants.

To solve the system of **first-order** linear differential equations:

y1' = y1 + 3y2

y2' = 3y1 - y2

We can rewrite the equations in **matrix **form:

Y' = AY

where Y = [y1, y2]^T is the vector of unknowns and A is the **coefficient matrix**:

A = [1, 3; 3, -1]

To find the solution, we need to diagonalize the coefficient matrix A.

The **eigenvalues** λ1 and λ2 can be found by solving the characteristic equation |A - λI| = 0, where I is the identity matrix:

|A - λI| = |1 - λ, 3; 3, -1 - λ| = (1 - λ)(-1 - λ) - 3(3) = λ^2 - 2λ - 10 = 0

Solving the quadratic equation, we find two distinct eigenvalues: λ1 = 4 and λ2 = -2.

Next, we find the corresponding eigenvectors by solving the systems (A - λ1I)v1 = 0 and (A - λ2I)v2 = 0:

For λ1 = 4:

(1 - 4)v1 + 3v2 = 0

-3v1 - 5v2 = 0

Solving this system, we find v1 = [3, -1]^T.

For λ2 = -2:

(1 + 2)v2 + 3v2 = 0

3v1 + v2 = 0

Solving this system, we find v2 = [-1, 3]^T.

The general solution of the system is given by:

Y = c1e^(λ1t)v1 + c2e^(λ2t)v2

Substituting the values, we have:

y1 = 3c1e^(4t) - c2e^(-2t)

y2 = -c1e^(4t) + 3c2e^(-2t)

where c1 and c2 are arbitrary constants.

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Jalen solved the equation lx-2l-1>2 by graphing. Which graph can Jalen use to show his solution?

### Answers

Jalen can use a number line** graph** to show the solution to the equation |x - 2| - 1 > 2. A number line graph is a line that represents the set of real numbers with corresponding points marked on it.

To graphically represent the **solution**, Jalen can start by marking the point 2 on the number line, as it is the value inside the absolute value expression.Next, Jalen can plot points on the number line to represent the values hat satisfy the inequality. Since |x - 2| - 1 > 2, the solution lies to the left and right of the point 2.

Jalen can shade the **regions** on the number line that satisfy the inequality. In this case, the shaded region will be to the left of 2 and to the right of 2, excluding the point 2.

This graphically represents the solution to the **equation**, demonstrating the range of values of x that make the inequality true.

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Find all the real square roots of each number. -1/121

### Answers

The real **square root **of -1/121 is -1/11.

To find the real square roots of a number, we need to find the **numbers **that, when squared, give us the original number. In this case, we want to find the real square roots of -1/121.

The real square roots of a number are the numbers that, when squared, give us the **original number**.

Step 1: Simplify the fraction.

-1/121 cannot be **simplified **further.

Step 2: Take the square root.

The square root of -1/121 is the number that, when squared, equals -1/121.

Step 3: Rewrite the fraction.

We can write -1/121 as (-1/11)^2.

Step 4: Take the square root.

The square root of (-1/11)^2 is -1/11.

The real square root of -1/121 is -1/11.

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8. (10 points) Solve the initial value problem y' + 2zy² = 0, y(1) = 1.

### Answers

The solution of the initial value problem y' + 2zy² = 0, y(1) = 1 is y = 1 / (2zx + 1 - 2z) with the **domain** z ≠ 1 / 2x.

Given: y' + 2zy² = 0 and y(1) = 1

The given differential equation is a first-order linear **differential equation** and can be solved using the method of separation of variables. So, let's start solving it.

y' + 2zy² = 0 ...(1)

Separating **variables**, we have:

dy / dx = - 2zy²

Integrating both sides, we have:

∫ 1 / y² dy = ∫ -2z dx

- 1 / y = -2zx + C ...(2)

where C is the constant of **integration**.

Now, applying the initial condition y(1) = 1 in equation (2), we get:

- 1 / 1 = -2z(1) + C

- 1 = -2z + C

C = -1 + 2z

Putting this value of C in equation (2), we get the solution of the given initial **value** problem as:

- 1 / y = -2zx - 1 + 2z

1 / y = 2zx + 1 - 2z

y = 1 / (2zx + 1 - 2z)

Thus, the solution of the initial value problem y' + 2zy² = 0, y(1) = 1 is y = 1 / (2zx + 1 - 2z) with the domain z ≠ 1 / 2x.

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A heart-shaped graph in a 2D plane is defined by the equation (x² + y² - 1)³= 4x²y³ Determine dy/dx by implicit differentiation. y = 6x(x² + y² - 1)² - 8xy¹ 6x(x² + y²-1)² + 8xy² 12x²y²-6y(x² + y²-1) 6x²2²-3x² + y²²-1² 3x(x² + y²-12²-4x¹ . 3y (x²+²-1²-2x²y

### Answers

The final **expression** for **differential** of dy/dx is given by (4x³ - 4xy² - 6x(x² + y² - 1)) / (3(x² + y² - 1)² - 4y²).

Answer: We are given the **equation **(x² + y² - 1)³= 4x²y³and we are supposed to determine dy/dx by implicit differentiation.

First, we differentiate the given equation with respect to x using the chain **rule** (since the equation is in the form of f(g(x)), we get:

(3x² + 3y² - 2) * 2(x² + y² - 1) * 2x + (3x² + 3y² - 2)³ * 2y * dy/dx

= 8xy³ + 4x² * 3y² * dy/dx

Now, we can rearrange the equation to solve for dy/dx, which gives:

dy/dx = (4x³ - 4xy² - 6x(x² + y² - 1)) / (3(x² + y² - 1)² - 4y²)

Therefore, we have: dy/dx = (4x³ - 4xy² - 6x(x² + y² - 1)) / (3(x² + y² - 1)² - 4y²)

Thus, the required **derivative** is (4x³ - 4xy² - 6x(x² + y² - 1)) / (3(x² + y² - 1)² - 4y²).

Conclusion:

Therefore, the final expression for dy/dx is given by (4x³ - 4xy² - 6x(x² + y² - 1)) / (3(x² + y² - 1)² - 4y²).

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If (α,β) is a saddle point of the function f(x,y)=11y^2 − y^3 +3x^2 −6xy, then α+β=

### Answers

If (,) is a saddle point of the **function **f(x,y)=21y^2−y^3+3x^2−6xy, then + can be either 0 or 30, depending on which saddle point is being referred to.

To find the value of α + β when (α, β) is a **saddle point** of the function f(x, y) = 21y^2 − y^3 + 3x^2 − 6xy, we need to determine the coordinates of the saddle point.

To find the saddle point, we need to find the critical points of the **function **where the partial derivatives with respect to x and y are both equal to zero.

Taking the partial derivative of f(x, y) with respect to x:

∂f/∂x = 6x - 6y

Setting it equal to zero:

6x - 6y = 0

Simplifying:

x - y = 0

x = y

Taking the partial **derivative **of f(x, y) with respect to y:

∂f/∂y = 42y - 3y^2 - 6x

Setting it equal to zero:

42y - 3y^2 - 6x = 0

Substituting x = y:

42y - 3y^2 - 6y = 0

Simplifying:

45y - 3y^2 = 0

3y(15 - y) = 0

From this equation, we have two possibilities:

1) y = 0

2) 15 - y = 0, which implies y = 15

For the first case, if y = 0, then x = 0 as well (from x = y). So, one critical point is (0, 0).

For the second case, if y = 15, then x = 15 as well (from x = y). So, another critical point is (15, 15).

Therefore, we have two critical points: (0, 0) and (15, 15).

The sum of the coordinates of these saddle points is:

α + β = 0 + 0 = 0 (for the point (0, 0))

α + β = 15 + 15 = 30 (for the point (15, 15))

So, depending on which saddle point is being referred to, the value of α + β can be either 0 or 30.

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Find the critical point and determine if the function is increasing or decreasing on the given intervals.

y=x5/2-5x2, x>0

Critical point: c =

The function is:

on (0, c).

on (c, [infinity]).

### Answers

The **critical points **for the **function **is (4. -70)

How to detemine the critical points for the function

From the question, we have the following parameters that can be used in our computation:

y = 5/2x - 5x²

The **critical points **are the points where the **derivative **of f(x) equals 0 or undefined when the **function **is defined

When f(x) is **differentiated**, we have

y' = 5/2 - 10x

Set to 0 and evaluate

10x = 5/2

So, we have

x = 2 * 10/5

Evaluate

x = 4

Recall that

y = 5/2x - 5x²

So, we have

y = 5/2(4) - 5(4)²

y = -70

Hence, the **critical points **for the **function **is (4. -70)

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Using the Chain Rule. Calculate dz/dt for the function. z = f(x, y) = (x + 3y - 2)², x = x(t) = 4 sin (4t), y = y(t) = 4 cos (2 t) and evaluate at t = π. Round your answer to two decimal

### Answers

[tex]Given the **function** `z = f(x, y) = (x + 3y - 2)²` where `x = x(t) = 4 sin (4t)`, `y = y(t) = 4 cos (2 t)`[/tex]First, we need to calculate the **partial** derivatives of `f` with respect to `x` and `y`.

Then, we can evaluate `dz/dt` using the** chain rule**.

[tex]We can compute `∂f/∂x` as:$$\frac{\partial}{\partial x} \left[(x + 3y - 2)^2\right] = 2(x + 3y - 2) \cdot \frac{\partial}{\partial x}(x + 3y - 2)$$$$= 2(x + 3y - 2) \cdot \frac{\partial}{\partial x} x = 2(x + 3y - 2) \cdot \cos (4t)$$[/tex]

[tex]Similarly, we can compute `∂f/∂y` as:$$\frac{\partial}{\partial y} \left[(x + 3y - 2)^2\right] = 2(x + 3y - 2) \cdot \frac{\partial}{\partial y}(x + 3y - 2)$$$$= 2(x + 3y - 2) \cdot \frac{\partial}{\partial y} y = 6(x + 3y - 2) \cdot \sin (2t)$$[/tex]

[tex]Using the chain rule, we can now find `dz/dt`:$$\frac{dz}{dt} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dt}$$$$= 2(x + 3y - 2) \cdot \cos (4t) \cdot 4\cos (4t) + 6(x + 3y - 2) \cdot \sin (2t) \cdot (-4\sin (2t))$$$$= 8(x + 3y - 2) \cdot \cos (4t) \cdot \cos (4t) - 24(x + 3y - 2) \cdot \sin (2t) \cdot \sin (2t)$$When `t = π`,[/tex]

[tex]we have:$$x(\pi) = 4 \sin (4\pi) = 0$$$$y(\pi) = 4 \cos (2\pi) = 4$$[/tex]

[tex]Therefore, we can evaluate `dz/dt` as:$$\frac{dz}{dt}\Biggr|_{t=\pi} = 8(x(\pi) + 3y(\pi) - 2) \cdot \cos (4\pi) \cdot \cos (4\pi) - 24(x(\pi) + 3y(\pi) - 2) \cdot \sin (2\pi) \cdot \sin (2\pi)$$$$= 8(4) \cdot (1) \cdot (1) - 24(14) \cdot (0) \cdot (0) = 128$$Therefore, `dz/dt` evaluated at `t = π` is equal to `128`[/tex].

[tex]**Rounding** this to two **decimal** places gives: `dz/dt ≈ 128.00`.[/tex]

Hence, the required value of dz/dt is 128.00.

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To **calculate **dz/dt for the function z = f(x, y) = (x + 3y - 2)², we need to apply the **chain rule**. We have:

dz/dt at t = π is -160.

x = x(t) = 4 sin(4t)

y = y(t) = 4 cos(2t)

Let's find the **partial derivatives** of f(x, y) with respect to x and y:

∂f/∂x = 2(x + 3y - 2)

∂f/∂y = 6(x + 3y - 2)

Now, we can find dz/dt using the chain rule:

dz/dt = (∂f/∂x) * (dx/dt) + (∂f/∂y) * (dy/dt)

Substituting the given values for x and y:

dx/dt = 4 * 4 cos(4t)

dy/dt = -4 * 2 sin(2t)

And substituting these derivatives and the partial derivatives of f(x, y) into the chain rule **equation:**

dz/dt = [2(4 sin(4t) + 3(4 cos(2t)) - 2)] * (4 * 4 cos(4t)) + [6(4 sin(4t) + 3(4 cos(2t)) - 2)] * (-4 * 2 sin(2t))

Simplifying this expression:

dz/dt = 32(4 sin(4t) + 3(4 cos(2t)) - 2) * cos(4t) - 48(4 sin(4t) + 3(4 cos(2t)) - 2) * sin(2t)

Now, **evaluate** dz/dt at t = π:

dz/dt = 32(4 sin(4π) + 3(4 cos(2π)) - 2) * cos(4π) - 48(4 sin(4π) + 3(4 cos(2π)) - 2) * sin(2π)

Simplifying further, sin(4π) = 0, cos(4π) = 1, cos(2π) = 1, and sin(2π) = 0:

dz/dt = 32(0 + 3(4) - 2) * 1 - 48(0 + 3(4) - 2) * 0

= 32(10) - 48(10)

= 320 - 480

= -160

Therefore, dz/dt at t = π is -160.

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5. [9 marks] Evaluate the following improper integral or show that it diverges: Lo X (x² - 1)2/3 dr.

### Answers

The given** improper integral **converges and its value is:

= 3/8 [[tex]8^{4/3} + 23[/tex]]

Now, We can evaluate the given improper integral:

⇒ [tex]\int\limits^3_0 {\frac{x}{(x^2 - 1)^{2/3} } } \, dx[/tex]

We can use the **substitution **u = x² - 1, which yields du/dx = 2x and dx = du/2x.

Substituting these expressions into the integral, we get:

⇒ [tex]\int\limits^3_0 {\frac{x}{(x^2 - 1)^{2/3} } } \, dx[/tex]

= [tex]\int\limits^8_0 {\frac{(u + 1)}{2u^{2/3} } } \, du[/tex] (using u = x² - 1)

Now, let's split the **integrand **into two fractions:

[tex]\int\limits^8_0 {\frac{(u + 1)}{2u^{2/3} } } \, du[/tex]= = [tex]\int\limits^8_0 {\frac{(u ^{1/3} )}{2} } \, du[/tex] + [tex]\int\limits^8_0 {\frac{1}{2u^{2/3} } } \, du[/tex]

We can evaluate the first **integral **using the power rule:

= [tex]\int\limits^8_0 {\frac{(u ^{1/3} )}{2} } \, du[/tex]= 3/2 [[tex]\frac{u^{4/3} }{4}[/tex]](0 to 8)

= [tex]\frac{3 (8^{4/3} - 1 ) }{8}[/tex]

For the **second integral,** note that 1/(2u^(1/3)) is a continuous, positive function on the interval [0, 8).

Therefore, the integral is proper and we can evaluate it using the limit definition of an improper integral:

[tex]\int\limits^8_0 {\frac{1}{2u^{2/3} } } \, du[/tex] = lim t->0+ ∫(t to 8) [tex]{\frac{1}{2u^{2/3} } } \, du[/tex]

= lim t->0+ [u^(2/3)]/3 | = 2/3 [[tex]8^{2/3} - t^{2/3}[/tex]]

Taking the limit as t **approaches **zero from the right, we get:

lim t->0+ 2/3 [[tex]8^{2/3} - t^{2/3}[/tex]] = 2/3 [[tex]8^{2/3} - 0^{2/3}[/tex]] = 2/3 [[tex]8^{2/3}[/tex]] = (2/3) (2²) = 8/3

Therefore, the given improper integral converges and its value is:

⇒ [tex]\int\limits^3_0 {\frac{x}{(x^2 - 1)^{2/3} } } \, dx[/tex] = [tex]\int\limits^8_0 {\frac{(u + 1)}{2u^{2/3} } } \, du[/tex]= [tex]\int\limits^8_0 {\frac{(u ^{1/3} )}{2} } \, du[/tex] + [tex]\int\limits^8_0 {\frac{1}{2u^{2/3} } } \, du[/tex]

= 3 [[[tex]8^{4/3} - 1][/tex] / 8 + 8/3

= 3/8 [[tex]8^{4/3} + 23[/tex]]

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Find an equation of the inverse relation. \[ y=8 x+6 \] A. \( x=\frac{y+6}{8} \) C. \( x=8 y-6 \) D. \( x=8 y+6 \)

### Answers

The given **equation** of relation is y = 8x + 6. To find an equation of the inverse **relation**, we need to swap x and y, and then solve for y.

Therefore, y = 8x + 6 ......(1)

Now we **interchange** x and y to get the **inverse** of the function x = 8y + 6.

Rearranging, we get;8y = (x - 6)

**Dividing** throughout by 8, we get; y = (x - 6)/8

Therefore, an equation of the inverse relation is given by;x = (y - 6)/8

Thus, the **option** (A) x = (y + 6)/8 is the **correct** answer.

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