Mathematics High School

## Answers

**Answer 1**

The **price-demand** equation is p(x) = -5.64x + 100

How to find the price-demand equation.

We can use the given information to set up a system of two **equations**.

The first equation is p(0) = 100, which tells us that when there is no **demand**, the price is $100 per unit.

The second equation is p(4) = 78.38, which tells us that when there is a demand of 4 units per week, the price is $78.38 per unit.

We can use these two equations to solve for the slope and y-intercept of the line that represents the relationship between **price** and demand.

Solving for the slope, we get dp/dx = (78.38 - 100)/(4 - 0) = -5.64.

Since we know that the marginal price dp/dx at x units of demand per week is **proportional** to the price p, we can write dp/dx = kp, where k is a constant of proportionality.

Substituting in our value for dp/dx and solving for k, we get k = -5.64/100 = -0.0564.

Now we can use the point-slope form of a linear equation to write the price-demand equation: p - 100 = -0.0564(x - 0), which simplifies to p(x) = -5.64x + 100.)

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## Related Questions

Solve parts a-c below about the following figures.

Two similar prisms with heights 4 cm and 7 cm.

a. Using the ratio of the smaller figure to the larger figure, what is their scale factor?

b. What is the ratio of their surface areas?

c. What is the ratio of their volumes?

Answers must be an integer or simplified fraction.

### Answers

The **ratio** of their volumes is 64/343.

We are given that;

The height of two prisms 4cm and 7cm

Now,

a) The scale factor of two similar prisms is the ratio of their corresponding side **lengths**. In this case, the scale factor is 4/7.

b) The ratio of their surface areas is equal to the square of their scale factor. Therefore, the ratio of their surface areas is (4/7)^2 = 16/49.

c) The ratio of their volumes is equal to the cube of their scale **factor**. Therefore, the ratio of their volumes is (4/7)^3 = 64/343.

Therefore, by the given **ratio** the answer will be 64/343.

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Mo needs to buy enough material to cover the walls around the stage for a theater performance. If he needs 79 feet of wall covering, what is the minimum number of yards of material he should purchase if the material is sold only by whole yards? Enter yMathour answer in the box.

### Answers

**Answer: Mo will need 26.3333333333 yards in order to cover the walls.**

**Step-by-step explanation:**

**One yard = 3 feet (36 inches)**

**You must find how many yards are in 79 feet.**

**79 feet/3 feet = 26.333...**

**Therefore, Mo needs ****26.333... yards**** of material if the material is only sold by yards.**

FILL IN THE BLANK. ∫ f(x) dx = (3x – 15)¹³ + C, find f(x). f(x)= _______

### Answers

The **derivative **of (3x – 15)¹³ + C with respect to x is simply f(x), so we have f(x) = d/dx [(3x – 15)¹³ + C].

To find f(x) given that ∫ f(x) dx = (3x – 15)¹³ + C, we need to differentiate both sides of the **equation. **

Using the** power rule **of differentiation to differenetiate the function, we can simplify this to f(x) = 39(3x – 15)¹². Therefore, f(x) = 117(x – 5)¹².

In summary, to find f(x) given an **indefinite integral **of f(x), we simply differentiate the expression with respect to x. In this case, f(x) is equal to 117(x – 5)¹².

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Using the definition of the derivative, find f(x). Then find f'(-2), f'(o), and f'(5) when the derivative exists. ) f(x) = 12 - х f'(x)=0 (Type an expression using x as the variable.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. f'(-2)= OB. The derivative does not exist. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. f(0) = B. The derivative does not exist. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. f'(5) = (Type an integer or a simplified fraction.) B. The derivative does not exist.

### Answers

The** derivative** of -x will be -1. Therefore, f'(x) =** -1.**

The given** function **is f(x) = 12 - x. To find the derivative of f(x), we need to differentiate the function with respect to x. We know that the derivative of a constant is zero. Hence, the derivative of -x will be -1. Therefore, f'(x) = -1.

We have to find f'(-2), f'(0), and f'(5) when the derivative exists. We can find f'(-2), f'(0), and f'(5) by substituting the **corresponding values** of x in the derivative function. f'(-2) = -1, f'(0) = -1, and f'(5) = -1.

Therefore, the correct options are:

f'(-2) = -1

f(0) = 12 and the derivative exists at x = 0

f'(5) = -1

The derivative exists at all points of the given function. Hence, the correct answers are:

option (A) f'(-2) = -1

option (A) f(0) = 12 and the derivative exists at x = 0

option (A) f'(5) = -1

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Find a polynomial with integer coefficients that has degree 3 and zeros - 4 and 31. Find in factored form first, and then expand. Please show all of your work!

### Answers

A **polynomial** with degree 3 and zeros -4 and 31 can be written as (x + 4)(x - 31)(x - 31), which expands to f(x) = x³ - 54x² + 787x - 1196.

A polynomial with **degree** 3 and zeros -4 and 31 can be written as (x + 4)(x - 31)(x - k), where k is an integer that can be determined by expanding the expression.

Since the polynomial has degree 3 and zeros -4 and 31, we can write it as:

f(x) = a(x + 4)(x - 31)(x - k)

where a is a constant that can be determined by considering the leading **coefficient** of the polynomial. Since the polynomial has degree 3, the leading coefficient must be a nonzero integer, and we can write:

f(x) = a(x + 4)(x - 31)(x - k)

To determine the value of k, we can use the fact that the polynomial has zeros -4 and 31. This means that f(-4) = f(31) = 0. Substituting these values into the expression for f(x), we get:

f(-4) = a(0)(-35)(-4 - k) = 0

f(31) = a(35)(0)(31 - k) = 0

Since these expressions are equal to 0, it follows that either a = 0 or (4 + k) = 0 and (31 - k) = 0. Since a must be nonzero, we have:

4 + k = 0 and 31 - k = 0

Solving these equations simultaneously, we get k = 31. Substituting this value of k into the expression for f(x), we get:

f(x) = a(x + 4)(x - 31)(x - 31)

**Expanding** this expression, we get:

f(x) = a(x² - 27x - 496)(x - 31)

Therefore, a polynomial with degree 3 and zeros -4 and 31 can be written as (x + 4)(x - 31)(x - 31), which expands to f(x) = x³ - 54x² + 787x - 1196.

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as the size of the sample increases, what happens to the shape of the sampling distribution of sample means? multiple choice it is negatively skewed. it is positively skewed. it cannot be predicted in advance. it approaches a normal distribution.

### Answers

The shape of the sampling **distribution **of sample means it approaches a **normal **distribution. **(option d).**

The **shape **of the sampling distribution of sample means is determined by the size of the sample. As the sample size increases, the shape of the distribution approaches a normal distribution. This is known as the Central Limit Theorem (CLT).

The CLT states that, regardless of the population's distribution, the sampling distribution of sample means will be approximately normal as the sample size increases. This means that even if the population is not normally distributed, the distribution of the **sample **means will still be normal as long as the sample size is large enough.

The reason for this is that as the sample size increases, the variability of the sample means decreases. This is because the sample means tend to cluster around the population mean, making the distribution more **concentrated **and less spread out. As a result, the distribution becomes more symmetrical and bell-shaped, resembling a normal distribution.

Therefore, the correct answer to the multiple-choice question is d) it approaches a normal distribution.

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What does the | (in prob(A|B)) stand for?

a. Stop

b. Start over

c. Ignore

d. Given

### Answers

The meaning of the **| symbol** for the** conditional probability **is given as follows:

**d.** Given.

What is Conditional Probability?

Conditional probability is the probability of one event happening, considering the result of a previous event. The **formula **is:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which:

P(B|A) is the probability of the event B happening, given that the event A happened.[tex]P(A \cap B)[/tex] is the probability of both the events A and B happening.P(A) is the probability of the event A happening.

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The function f(x) = x2 is translated 7 units to the left and 3 units down to form the function g(x). which represents g(x)? g(x) = (x âˆ’ 7)2 âˆ’ 3 g(x) = (x 7)2 âˆ’ 3 g(x) = (x âˆ’ 3)2 âˆ’ 7 g(x) = (x âˆ’ 3)2 7

### Answers

The function g(x) = (x + 7)^2 - 3 represents the **translation **of f(x) = x^2 7 units to the left and 3 units down.

The given **function **f(x) = x^2 is translated 7 units to the left and 3 units down to form the function g(x). The rule for a **horizontal translation** of a function by h units to the left is f(x+h), and the rule for a vertical translation of a function by k units downwards is f(x)-k.

Therefore, the rule for g(x) is g(x) = (x+7)^2 - 3, as the function is translated 7 units to the left and 3 units down. The term (x+7)^2 represents the horizontal translation of the function f(x) by 7 units to the left, and the -3 term represents the **vertical translation** of the function f(x) by 3 units downward.

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Given 2 = ln(x + In y), evaluate Zx, Zy.

### Answers

The partial **derivatives **Zx and Zy are:

Zx = 1/(x + ln y)

Zy = (1/y)/(x + ln y)

Given the equation 2 = ln(x + ln y), we need to evaluate the **partial **derivatives Zx and Zy with respect to x and y, respectively.

Differentiate the equation with respect to x.

For Zx, we treat y as a constant and differentiate the equation with respect to x:

Zx = d/dx[ln(x + ln y)].

Using the chain rule, we get:

Zx = 1/(x + ln y) * d/dx(x + ln y)4

Since the derivative of x with respect to x is 1 and the derivative of ln y with **respect **to x is 0 (as y is treated as a constant):

Zx = 1/(x + ln y)

Step 2: Differentiate the equation with respect to y.

For Zy, we treat x as a **constant **and differentiate the equation with respect to y:

Zy = d/dy[ln(x + ln y)]

Using the chain rule, we get:

Zy = 1/(x + ln y) * d/dy(x + ln y)

Since the derivative of x with respect to y is 0 (as x is treated as a constant) and the derivative of ln y with respect to y is 1/y:

Zy = 1/(x + ln y) * (1/y).

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A: The set of students who are computer science majors. B: The set of students who are taking CSE 191. Express the following sets in terms of A and B. Use set operators when necessary.

### Answers

The appropriate set operators (∩, ∪, -, ') when expressing these **sets** in terms of A and B.

How to use set operators ?The set of **computer science** majors who are taking CSE 191 can be represented as the intersection of A and B: A ∩ B.The set of students who are not computer science majors but are taking **CSE 191** can be represented as the difference of B and A: B - A.The set of **students** who are either computer science majors or taking CSE 191, or both can be represented as the union of A and B: A ∪ B.The set of students who are neither computer science majors nor taking CSE 191 can be represented as the complement of the union of A and B: (A ∪ B)'.

Remember to use the appropriate set operators (∩, ∪, -, ') when expressing these sets in terms of A and B.

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The height of a box containing a certain printer cartridge is 1 in more than the width and the length is 7 in. more than the width. If the volume of the box is 120 in.3, then what are the dimensions?

### Answers

Let's say the width of the box is "x".According to the question, the **height **of the box is 1 inch more than the width, so the height would be "x + 1".

Similarly, the length of the box is 7 inches more than the width, so the length would be "x + 7". The volume of the box is given as 120 in.3, so we can use the **formula **for the volume of a rectangular box: Volume = Length x Width x Height

Substituting the values we have: 120 = (x+7) x (x) x (x+1) .

Expanding this **equation**: 120 = x^3 + 8x^2 + 7x, Rearranging: x^3 + 8x^2 + 7x - 120 = 0, We can solve for "x" using trial and error, or by using a graphing calculator. After some calculations, we find that the width of the box is approximately 3.3 inches. Using this **value**, we can find the length and height of the box: Length = 3.3 + 7 = 10.3 inches, Height = 3.3 + 1 = 4.3 inches . So the dimensions of the box are: Length = 10.3 inches, Width = 3.3 inches, Height = 4.3 inches.

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What is the area of half the garden

### Answers

**Answer:**

**Step-by-step explanation:**

If the gardens diameter is 20 inches, then half of the diameter is 10. because half of 20 is 10.

please actually try on your problem before you make a post. or at least try to count.

True of False?

1. The Central Limit Theorem (CLT) states that the sample mean is normally distributed whenever the population size is sufficiently large, even if the underlying population distribution is skewed.

2. Suppose that a set of data has been collected over time. A time plot of the data set shows an increasing trend. This means that we have evidence that the observations are not iid (independent, identically distributed).

3. The U.S. Census Bureau reported in 2014 that the mean salary for statisticians was $96,000. A researcher speculates that the mean salary is too high for statisticians who have limited work experience (less than 2 years of work experience). To put this theory to the test, the researcher took a random sample of 45 statisticians who had limited work experience (less than 2 years of work experience) and recorded their 2014 annual salary. You have been asked to use the data to test (at a 10% level) the following hypotheses: H0: μ = 96,000 versus Ha: μ < 96,000.

The hypotheses involve the parameter μ. Is this definition for the parameter correct or incorrect?

"μ = the population mean salary for all statisticians reported by the U.S. Census Bureau in 2014."

### Answers

1. The given statement is **False.**

2. The given statement is False.

3. The given statement is True.

False. The CLT states that the sample mean approaches a normal distribution as the sample size increases,** regardless** of the population distribution being sampled from.

False. An **increasing trend** in a time plot suggests that there is a relationship between the observations, but it does not necessarily imply that they are not iid. Additional tests or analyses would be needed to confirm or refute independence.

Correct. μ is defined as the **population means salary** for statisticians who have limited work experience, which is the parameter being tested in the hypotheses.

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хп Consider the power series In(n + 4) Find the radius of convergence, R. (If R=0, type "inf".) R . Answer: R= Find the interval of convergence. Answer in interval notation. Answer: 10

### Answers

The power series In(n+4) has a radius of convergence of **0** and no interval of **convergence**, meaning it diverges for all values of **n. **

To determine the radius of convergence, we can use the ratio test, which involves taking the limit of the ratio of consecutive terms as n approaches infinity. If this **limit **exists and is less than 1, the series converges absolutely, and if it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and we must use a different method.

Applying the ratio test to In(n+4), we get:

lim as n approaches infinity of |(n+1+4)/(n+4)| = 1

If this limit exists and is less than 1, the series converges absolutely, and if it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive.

Applying the root test to In(n+4), we get:

lim as n approaches **infinity **of |(n+4)^(1/n)| = 1

Since the limit is exactly 1, the root test is inconclusive as well. However, we can use the fact that the natural logarithm function is increasing to rewrite the series as:

In(n+4) > In(n)

Thus, we can compare the given series to the harmonic series, which is a divergent series of the form 1/n. Since In(n) diverges as n approaches infinity, the given series must also **diverge **for all values of n. Therefore, the radius of convergence is 0.

As for the interval of convergence, since the series diverges for all values of n, there is no interval of convergence. We can write this in interval notation as (-∞, ∞).

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Given the function z(x, y) = 5x^2 – 3y^2 – 30x + 7y + 4xy, what are the critical values (x,y), according to the FOC?

### Answers

The **critical values** of the function z(x, y) = 5x² – 3y² – 30x + 7y + 4xy, according to the FOC, is (2, 5/2).

The FOC states that "At the highest and lowest points of a curve, the tangent to the curve at such points is horizontal. The slope of the curve is zero."

To find the critical values of z(x,y) according to the FOC (first-order conditions), we need to take partial derivatives of the **function** z(x, y) = 5x² – 3y² – 30x + 7y + 4xy with respect to x and y and set them equal to zero:

∂z/∂x = 10x - 30 + 4y = 0

∂z/∂y = -6y + 7 + 4x = 0

Solve the equation 10x - 30 + 4y = 0 for x:

[tex]\begin{aligned}& 10 x+4 y-30+30=0+30\\& 10 x+4 y=30 \\& 10 x+4 y+-4 y=30+-4 y \\& 10 x=-4 y+30 \\& \frac{10 x}{10}=\frac{-4 y+30}{10} \\& x=\frac{-2}{5} y+3\end{aligned}[/tex]

Substitute [tex]$\frac{-2}{5} y+3$[/tex] for x in 4 x-6 y+7=0:

[tex]\begin{aligned}& 4\left(\frac{-2}{5} y+3\right)-6 y+7=0 \\& \frac{-38}{5} y+19=0 \\& \frac{-38}{5} y+19+-19=0+-19 \\& \frac{-38}{5} y=-19 \\& \left.\frac{\frac{-38}{5} y}{\frac{-38}{5}}=\frac{-19}{\frac{-38}{5}} \\& y=\frac{5}{2}\end{aligned}[/tex]

Substitute 5/2 for y in [tex]x=\frac{-2}{5} y+3[/tex]:

[tex]$$\begin{aligned}& x=\frac{-2}{5} y+3 \\& x=\frac{-2}{5}\left(\frac{5}{2}\right)+3\end{aligned}$$$x=2$[/tex]

So, we get:

x = 2

y = 5/2

Therefore, the **critical values** (x,y) are (2, 5/2).

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2. Write the first five terms of the following sequences (start with n=1); a (-1)"+13 2n+1 b. by 3. Check convergence for the following sequences: a. an = 1+(-3) b. ant 2n 7+1

### Answers

Let's break it down into parts:

1. Write the first five terms of the following sequences:

a. an = (-1)^n + 13

The first five terms for this** sequence** are:

a1 = (-1)^1 + 13 = -1 + 13 = 12

a2 = (-1)^2 + 13 = 1 + 13 = 14

a3 = (-1)^3 + 13 = -1 + 13 = 12

a4 = (-1)^4 + 13 = 1 + 13 = 14

a5 = (-1)^5 + 13 = -1 + 13 = 12

b. bn = 2n + 1

The first five** terms** for this sequence are:

b1 = 2(1) + 1 = 2 + 1 = 3

b2 = 2(2) + 1 = 4 + 1 = 5

b3 = 2(3) + 1 = 6 + 1 = 7

b4 = 2(4) + 1 = 8 + 1 = 9

b5 = 2(5) + 1 = 10 + 1 = 11

2. Check **convergence** for the following sequences:

a. an = 1 + (-3)^n

This sequence does not converge, as the term (-3)^n will **oscillate** between positive and negative values without approaching a specific limit.

b. I believe there is a typo in the provided sequence "ant 2n 7+1". Please provide the correct sequence, and I'll be happy to help you check its convergence.

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1)estimate Δf=f(6.1)−f(6) when f(x)=x^4

Δf≈

be sure to find the estimate not the actual

### Answers

The **estimated** Δf is approximately 86.4.

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point.

To estimate Δf = f(6.1) - f(6) when f(x) = x⁴, we can use the **tangent** line approximation method.

First, we need to find the derivative of f(x).

Finding the derivative of f(x) using the power rule of differentiation:

f'(x) = 4x³

Calculate f'(6) to get the following:

f'(6) = 4(6³)

= 4(216) = 864

Calculate the **estimated change** which is given by:

Δf ≈ f'(6)(6.1-6) = 864(0.1) = 86.4

So, the estimated Δf is approximately 86.4. Note that this is an estimate and not the actual value.

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"Use the Integral Test to determine whether the infinite series is convergent.

∑n^n^2/(2n+1)^n

### Answers

This **integral **converges since the integrand is a **Gaussian **function with a negative exponent, and the limits of integration are from 1 to infinity. Therefore, by the Integral Test, the series ∑n²(n²)/(2n+1)²n converges.

In mathematics, the integral test is a method used to test whether the monotonic term of an infinite function is convergent. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes referred to as the Maclaurin-Cauchy test.

To use the Integral Test to determine the convergence of the series ∑n²(n²)/(2n+1)²n, we need to compare it with an integral.

Let f(x) = x²(x²)/(2x+1)²x. Then, the Integral Test states that if the improper integral ∫f(x)dx from 1 to infinity converges, then the series ∑f(n) also converges. If the integral diverges, then the series also diverges.

We have:

∫f(x)dx = ∫x²(x²)/(2x+1)²xdx

To simplify the integral, we use u-substitution with u = x², so that du/dx = 2x and dx = du/(2x). Substituting, we get:

∫f(x)dx = ∫u²(u)/(2√u+1)²(u/2) ×du/(2√u)

Now, we can simplify the integral using the fact that:

(2√u+1)²(u/2) > 2²

So, we get:

∫f(x)dx < ∫u²(u)/(2² ×2√u)du

Using the fact that u²(u) < e²(u²), we have:

∫f(x)dx < ∫e²(u²-u×log(2)-1/2)du

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Evaluate the indefinite integral ∫7 sec2(x) dx. (Use C for the constant of integration.)

### Answers

The **indefinite integral** of ∫7 sec²(x) dx is :

** **7 tan(x) + C

where C is the constant of integration.

To calculate the** indefinite integral **of ∫7 sec²(x) dx follow the following steps :

Step 1: Identify the **function** to integrate, which is 7 sec²(x) in this case.

Step 2: Recall that the integral of sec²(x) dx is **tan(x)**, so we will apply this knowledge to the given function.

Step 3: Multiply the integral of sec²(x) by the constant 7, which is 7 tan(x).

Step 4: Add the constant of integration, C.

Therefore, the indefinite integral of ∫7 sec²(x) dx is **7 tan(x) + C**, where C is the constant of integration.

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Solve

y" (t) + 4y' (t) + 5y(t) =-2 cosht

### Answers

The** general solution** to the differential equation y" (t) + 4y' (t) + 5y(t) = -2 cosh(t) is y(t) = e^(-2t) (c1 cos(t) + c2 sin(t)) - (1/2) cosh(t) + (2/3) sinh(t), where c1 and c2 are** constants** determined by the initial conditions.

To solve the **differential equation**

y" (t) + 4y' (t) + 5y(t) = -2 cosh(t)

We first find the characteristic equation

r^2 + 4r + 5 = 0

Using the quadratic formula, we find that the roots are

r = (-4 ± sqrt(4^2 - 4(5)))/2 = -2 ± i

Therefore, the general solution to the **hom*ogeneous equation**

y" (t) + 4y' (t) + 5y(t) = 0

is given by

y_h(t) = e^(-2t) (c1 cos(t) + c2 sin(t))

To find a particular solution, we will use the method of undetermined coefficients and assume a particular solution of the form

y_p(t) = A cosh(t) + B sinh(t)

Taking the first and second derivatives of y_p(t), we get

y_p'(t) = A sinh(t) + B cosh(t)

y_p''(t) = A cosh(t) + B sinh(t)

Substituting these into the **original differential** equation, we get

(A cosh(t) + B sinh(t)) + 4(A sinh(t) + B cosh(t)) + 5(A cosh(t) + B sinh(t)) = -2 cosh(t)

Simplifying this equation, we get

(6A + 4B) cosh(t) + (4A + 6B) sinh(t) = -2 cosh(t)

Equating the coefficients of cosh(t) and sinh(t) on both sides, we get the system of equations

6A + 4B = -2

4A + 6B = 0

Solving this system, we get

A = -1/2

B = 2/3

Therefore, a **particular solution** to the differential equation is

y_p(t) = (-1/2) cosh(t) + (2/3) sinh(t)

The general solution to the differential equation is then given by the sum of the hom*ogeneous and particular solutions

y(t) = y_h(t) + y_p(t) = e^(-2t) (c1 cos(t) + c2 sin(t)) - (1/2) cosh(t) + (2/3) sinh(t)

where c1 and c2 are constants determined by the **initial conditions.**

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The volume of a rectangular prism is 50 5/8 cubic inches.The dimensions are: Length 5 in, width ? , and Height 2 1/4 . This is 6 th grade math btw

### Answers

**Answer:**

9/2 or 4 1/2

**Step-by-step explanation:**

50 5/8 / 2 1/4 * 5

50 5/8 / 11 1/4

9/2 or 4 1/2

Find the volume of the solid obtained by rotating the region bounded by y= 32x^2, x = 1, y = 0, about the z-axis. Answer:

### Answers

**Volume **of the **solid **obtained by rotating the region bounded by y= 32x², x = 1, y = 0, about the z-axis is π/3 cubic units.

What is indetail solution to find volume of the solid obtained by rotating the region?

To find the volume of the solid obtained by rotating the region bounded by y= 32x², x = 1, y = 0, about the **z-axis**, we can use the formula:

V = π∫[a,b] (f(x))² dx

Where f(x) is the distance between the **curve **and the axis of rotation (in this case, the z-axis) and [a,b] is the interval of **integration**.

First, we need to find f(x), which is simply the value of x. This is because the distance between the curve y= 32x² and the z-axis is equal to the **distance **between the point (x, 0) on the curve and the z-axis, which is just x.

Next, we need to find the **limits of integration**. Since x = 1 is the right boundary of the region, we can integrate from 0 to 1.

Therefore, the volume of the solid is:

V = π∫[0,1] (x)² dx

= π[x³/³] from 0 to 1

= π/3

So the volume of the solid obtained by **rotating the region **bounded by y= 32x², x = 1, y = 0, about the z-axis is π/3 cubic units.

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The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs is given by the following function C(x)= 2000 + 2x - 0.0001x2 (0 ≤ x ≤ 6000) (a) What is the actual cost incurred in producing the 1021st and the 2001st disc? (Round your answers to the nearest cent.) 1021st disc $ =2001st disc $ =(b) what is the marginal cost when x = 1020 and 2000? (Round your answers to the nearest cent) 1020 $=2000 $=

### Answers

The **Marginal **cost when x=1020 is $1.96.

When x=2000:

C'(2000) = 2 - 0.0002(2000)

C'(2000) = $0

(a) To find the cost **incurred** in producing the 1021st disc, we need to substitute x=1021 into the given function:

C(1021) = 2000 + 2(1021) - 0.0001(1021)^2

C(1021) = 2000 + 2042 - 104.2441

C(1021) = $2937.76

So the cost incurred in **producing** the 1021st disc is $2937.76.

Similarly, to find the cost incurred in producing the 2001st disc, we need to **substitute** x=2001 into the given function:

C(2001) = 2000 + 2(2001) - 0.0001(2001)^2

C(2001) = 2000 + 4002 - 400.8001

C(2001) = $5597.20

So the cost incurred in producing the 2001st disc is $5597.20.

(b) The marginal cost is the **derivative** of the total cost function C(x) with respect to x. So we need to find C'(x) and **evaluate** it at x=1020 and x=2000.

C(x) = 2000 + 2x - 0.0001x^2

C'(x) = 2 - 0.0002x

When x=1020:

C'(1020) = 2 - 0.0002(1020)

C'(1020) = $1.96

So the marginal cost when x=1020 is $1.96.

When x=2000:

C'(2000) = 2 - 0.0002(2000)

C'(2000) = $0

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Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, R(X), and cost. C(x) of producing x units are in dollars. R(x)=60x-0.1x^2 C(x) = 4x + 30 In order to yield the maximum profit of $ ____ . ____ units must be produced and sold. (Simplify your answers. Round to the nearest cent as needed.)

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The **maximum profit** and the **number** of **units** that must be **produced** completes the statement; In order to yield the maximum profit of $7,870, 280 units must be produced and sold.

What is a profit?

A **profit** is the difference between the **revenues** **and** **costs** of a **business**.

The **profit** function is; R(x) = 60·x - 0.1·x²

The **cost** function is; C(x) = 4·x + 30

The profit **function** is; P(x) = R(x) - C(x)

Therefore; P(x) = 60·x - 0.1·x² - 4·x + 30 = 56·x - 0.1·x² + 30

P(x) = 56·x - 0.1·x² + 30 = -0.1·x² + 56·x + 30

The **maximum** **profit** **value** is the value of the profit function at the vertex point of the **parabola** representing the graph of the** profit function**, which can be found as follows;

The maximum point is the point at which; *x* = -b/(2·a)

Where; b = 56, a = (-0.1), we get;

*x* = -56/(2 × (-0.1)) = 280

The maximum profit is therefore; P(280) = -0.1 × 280² + 56 × 280 + 30 = 7870

Therefore; In order to yield maximum profit, $7,870, 280 units must be produced and sold

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Determine where the given function is concave up and where it is concave down. 2 f(x) = x4 – 24x? = X A. Concave up on (-00, -213) and (213,00), concave down on (-213, 213) B. Concave up on (-0, -2) and (0, 2), concave down on (-2, 0) and (2, 0) C. Concave up on (-0, - 2) and (2, oo), concave down on (-2, 2) D. Concave down on (-[infinity], - 2) and (2, 0), concave up on (-2,2)

### Answers

The correct answer is A. The function is** concave** up on** (-∞, -2) and (2, ∞),** and** concave down** on **(-2, 2).**

The** second derivative** of the given function is f''(x) = 12x^2 - 24. To determine where the function is concave up or down, we need to find the roots of the second derivative and test the intervals between them.

Setting f''(x) = 0, we get 12x^2 - 24 = 0, which gives x = ±2. Therefore, the critical points are x = -2 and x = 2.

Testing the interval (-∞, -2), we pick a value x = -3 and plug it into f''(x) = 12x^2 - 24. We get f''(-3) = 84, which is positive. Therefore, the function is concave up on (-∞, -2).

Testing the interval (-2, 2), we pick a value x = 0 and plug it into f''(x) = 12x^2 - 24. We get f''(0) = -24, which is negative. Therefore, the function is concave down on (-2, 2).

Testing the interval (2, ∞), we pick a value x = 3 and plug it into f''(x) = 12x^2 - 24. We get f''(3) = 84, which is positive. Therefore, the function is concave up on (2, ∞).

Therefore, the correct answer is A. The function is** concave** up on** (-∞, -2) and (2, ∞),** and** concave down** on **(-2, 2).**

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Simplify the expression by combining like terms

(-6x + 2) - (3x + 4)

### Answers

simplified expression is -9x - 2.

The simplified answer is -9x-2

Simplify the exponential expression. a) (-8x^5 y)(-10x^2 y^6)

### Answers

simplified **exponential expression** (-8x^5 y)(-10x^2 y^6) is: 80x^7 y^7.

To simplify the exponential expression (-8x^5 y)(-10x^2 y^6), we can first multiply the **constants **(-8 and -10) to get 80. Then we can **multiply **the **variables **with the same base (x and y) by adding their exponents. So, x^5 multiplied by x^2 gives us x^7, and y multiplied by y^6 gives us y^7.

Therefore, the simplified exponential expression is 80x^7 y^7.

simplify the exponential expression. Given the expression (-8x^5 y)(-10x^2 y^6), we can simplify it by multiplying the **coefficients **and adding the **exponents **of like terms.

Your simplified expression is: 80x^7 y^7.

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Solve the following Exact/Inexact Differential Equations. If itis inexact, then solve it by finding the Integrating Factor. (4t 3y − 15t 2 − y) dt + (t 4 + 3y 2 − t) dy = 0

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The given differential equation is: [tex](4t 3y − 15t 2 − y) dt + (t 4 + 3y 2 − t) dy = 0[/tex] Taking the **partial derivative **of F with respect to y and equating it to the coefficient of dy, we get: [tex]∂F/∂y = e^(t^2/2 - 5t[/tex]

[tex]∂/∂y (4t 3y − 15t 2 − y) = 12t y - 1 ∂/∂t (t 4 + 3y 2 − t) = 4t 3 - 1[/tex] Since these partial derivatives are not equal, the given differential equation is not exact. To solve this equation, we need to find the **integrating factor**.

The integrating factor is a function that we can multiply both sides of the **differential equation** by, which will make it exact. Let's assume the integrating factor is [tex]µ(t, y)[/tex], then [tex]µ(t, y) * (4t 3y − 15t 2 − y) dt + µ(t, y) * (t 4 + 3y 2 − t) dy = 0[/tex]

After some **simplification**, we get: [tex]µ(t, y) = e^(t^2/2 - 5t + 3y[/tex]) Multiplying both sides of the differential equation by this integrating factor, we get: [tex]e^(t^2/2 - 5t + 3y) * (4t 3y − 15t 2 − y) dt + e^(t^2/2 - 5t + 3y) * (t 4 + 3y 2 − t) dy = 0[/tex]

This **equation **is now exact, and we can solve it by finding its potential function. The potential function is the function whose partial derivatives give the coefficients of the differential equation.

Integrating partial derivative with t gives the **potential function**: [tex]F(t, y) = ∫ e^(t^2/2 - 5t + 3y) * (4t 3y − 15t 2 − y) dt = e^(t^2/2 - 5t + 3y) * (t^4/4 - 5t^3/3 - ty) + g(y)[/tex]

Taking the partial derivative of F with respect to y and equating it to the **coefficient **of dy, we get: [tex]∂F/∂y = e^(t^2/2 - 5t[/tex]

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if the chi-squared goodness-of-fit test is used with a significance of 0.05 to test whether random values have been appropriately generated by the device, what is the p-value of the test rounded to two places after the decimal, and what is the appropriate decision?

### Answers

The** random number** generation by the device is not appropriate for generating values according to the** Binomial distribution** with n=5 and p=0.45.

To test whether the generated values follow the expected Binomial distribution, we can use the** Chi-squared goodness-of-fit test**. The steps to perform this test are as follows

Define the null hypothesis and alternative hypothesis. In this case, the **null hypothesis** is that the generated values follow the expected Binomial distribution with n=5 and p=0.45. The alternative hypothesis is that the generated values do not follow this distribution.

Choose a significance level. In this case, the significance level is 0.05.

Calculate the expected frequencies for each category of the Binomial distribution with n=5 and p=0.45. We can use the formula for the Binomial distribution to calculate the **probabilities**, and then multiply them by the total number of observations to get the expected frequencies.

Expected frequency for each category = P(category) x total number of observations

Expected frequency for category 0 = P(0) x 1200 = 0.0176 x 1200 = 21.12

Expected frequency for category 1 = P(1) x 1200 = 0.1284 x 1200 = 154.08

**Expected frequency** for category 2 = P(2) x 1200 = 0.3574 x 1200 = 428.88

Expected frequency for category 3 = P(3) x 1200 = 0.4162 x 1200 = 499.44

Expected frequency for category 4 = P(4) x 1200 = 0.1949 x 1200 = 233.88

Expected frequency for category 5 = P(5) x 1200 = 0.0055 x 1200 = 6.6

Calculate the Chi-squared test statistic. The formula for the Chi-squared test statistic is:

**Chi-squared** = Σ ( (observed frequency - expected frequency)^2 / expected frequency )

where Σ is the sum over all categories. Using the expected frequencies and the observed frequencies from the table, we can calculate the Chi-squared test statistic

Chi-squared = ( (28-21.12)^2 / 21.12 ) + ( (168-154.08)^2 / 154.08 ) + ( (423-428.88)^2 / 428.88 ) + ( (459-499.44)^2 / 499.44 ) + ( (105-233.88)^2 / 233.88 ) + ( (17-6.6)^2 / 6.6 ) = 94.67

Calculate the** degrees of freedom**. The degrees of freedom for the Chi-squared test are equal to the number of categories minus 1. In this case, there are 6 categories, so the degrees of freedom are 5.

Calculate the P-value of the test. We can use a Chi-squared distribution table or a calculator to find the P-value for the Chi-squared test with 5 degrees of freedom and a test statistic of 94.67. The **P-value** is less than 0.01 (about 0.000000000000000000000000000000000000000001), so we reject the null hypothesis that the generated values follow the expected Binomial distribution with n=5 and p=0.45.

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The given question is incomplete, the complete question is:

A random number generation device is expected to generate random values according to the Binomial distribution with n=5 and p=0.45. To ensure that the device's random number generation is appropriate, 1200 recently generated random values by this device have been organized in the following table. If the Chi-squared goodness-of-fit test is used with a significance of 0.05 to test whether random values have been appropriately generated by the device, what is the P-value of the test rounded to two places after the decimal, and the decision made?

What is the amplitude of the sinusoidal function shown?

### Answers

The **amplitude** of the graph of a **sine function** is 2.

Given is **sinusoidal function**, we need to find the **amplitude** of the function.

We know,

The **amplitude** of the graph of a sine function is the **vertical distance** from the top of a peak to the center line.

This is the same as the vertical distance from the top of a peak to the lowest point on the graph, divided by 2.

The vertical distance = 8

Amplitude = 8

Hence the **amplitude** of the graph of a **sine function** is 2.

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