The **r** in **math example** is a real number .

**What is an integer ?**

integer can be defined as a whole number but **not a fraction** number in which it can be** positive , negative or zero** and integer is denoted by the **letter Z** , integers additionally does now not include the fractional part. for that reason, we are able to say, integers are numbers that can be positive, negative or 0, but cannot be a fraction. we will carry out all the arithmetic **operations**, like addition, subtraction, multiplication and division, on integers.

The examples of** integers **are, 1, 2,-4,-8 and etc.

**Real number **can be any** number**, fraction, decimal, whole, positive, negative

So the **examples **for the real numbers are** 1/2 , 2.3 , +4 , -69 **and etc .

Hence, The **r** in math example is a **real number . **

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a test statistic value of 2.14 puts it in the rejection region. if the test statistic is actually 2.19 then we know the p-value is less than the significance level for the test. true or false

The statement is** True.**

A test statistic value of 2.14 puts it in the rejection region, which means that if the **null hypothesis** is true, the probability of obtaining a test statistic as extreme as 2.14 or more extreme is less than the significance level of the test. Therefore, we reject the null hypothesis at the given significance level.

If the test statistic is actually 2.19, which is more extreme than 2.14, then the probability of obtaining a test statistic as extreme as 2.19 or more extreme under the **null hypothesis** is even smaller than the probability corresponding to a test statistic of 2.14.

This means that the p-value for the test is even smaller than the significance level, and we reject the **null hypothesis** with even greater confidence.

In other words, if the test statistic is more extreme than the critical value, the p-value is smaller than the significance level, and we reject the** null hypothesis **at the given significance level with greater confidence.

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The scale on a map of Fort Landon is 5 inches = 95 miles. If the length on the map between Snake World and the International Space Center measures 4 inches, what is the actual distance in miles?

the actual **distance** between Snake World and the **International** Space Center is 76 miles.

To find the actual distance in miles between Snake World and the International Space Center, we need to use the given scale of the map: 5 inches = 95 miles.

If 5 inches on the map represents 95 miles, we can set up a proportion to find the actual distance in miles for the **measured** length on the map.

Let's **denote** the actual distance in miles as "x".

According to the given scale, we have the proportion:

5 inches / 95 miles = 4 inches / x miles

We can cross-multiply to solve for x:

5 inches * x miles = 4 **inches** * 95 miles

Simplifying further:

5x = 380

Dividing both sides by 5:

x = 380 / 5

Calculating the value:

x = 76

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exercise 6.1.11: find the inverse laplace transform of 1 (s−1) 2 (s 1) .

The **inverse Laplace** transform of 1/((s-1)^2 (s+1)) is (1/4)e^t - (1/2)te^t + (1/4)e^(-t).

To find the inverse Laplace transform of the given function:

F(s) = 1 / ((s-1)^2 (s+1))

We can use **partial fraction** decomposition to break it down into simpler terms:

F(s) = A / (s-1) + B / (s-1)^2 + C / (s+1)

To solve for the coefficients A, B, and C, we can multiply both sides of the equation by the **denominator** and substitute in values of s to obtain a system of linear equations. After solving for A, B, and C, we get:

A = 1/4, B = -1/2, and C = 1/4

Now, we can use the inverse Laplace transform formulas to obtain the time **domain** function:

f(t) = (1/4)e^t - (1/2)te^t + (1/4)e^(-t)

Therefore, the inverse Laplace transform of 1/((s-1)^2 (s+1)) is (1/4)e^t - (1/2)te^t + (1/4)e^(-t).

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The** inverse Laplace transform** of 1/(s-1)^2(s+1) is 1/2 e^t + 1/2 t e^t - 1/4 e^-t.

The inverse Laplace transform of 1/(s-1)^2(s+1) is:

f(t) = L^-1 {1/(s-1)^2(s+1)}

Using** partial fraction** decomposition:

1/(s-1)^2(s+1) = A/(s-1) + B/(s-1)^2 + C/(s+1)

Multiplying both sides by (s-1)^2(s+1), we get:

1 = A(s-1)(s+1) + B(s+1) + C(s-1)^2

Substituting s=1, we get:

1 = 2B

B = 1/2

Substituting s=-1, we get:

1 = 4C

C = 1/4

Substituting B and C back into the** equation**, we get:

1/(s-1)^2(s+1) = 1/(2(s-1)) + 1/(2(s-1)^2) - 1/(4(s+1))

Taking the inverse Laplace transform of each term, we get:

f(t) = L^-1 {1/(2(s-1))} + L^-1 {1/(2(s-1)^2)} - L^-1 {1/(4(s+1))}

Using the Laplace transform table, we get:

f(t) = 1/2 e^t + 1/2 t e^t - 1/4 e^-t

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Roberto compró 6 cd's y 10 revistas en $ 900.00 pesos; en la misma tienda su amiga María compró 10 cd's y 4

revistas en $ 1.220.00 pesos. ¿ Cual es el sistema de ecuaciones con dos incognitas que representa el problema?

The **system of linear equation** that represent this problem is

6x + 10y = 900

10x + 4y = 1220

What is the system of equation?Let's represent the number of CDs Roberto bought as x and the number of magazines as y

The problem states the following information:

Using the** variables**; x and y as given;

1. Roberto bought 6 CDs and 10 magazines for $900.00 pesos. This can be represented as the** equation**:

6x + 10y = 900

2. María bought 10 CDs and 4 magazines for $1,220.00 pesos. This can be represented as the equation:

10x + 4y = 1220

So, the **system of equations** representing the problem is:

6x + 10y = 900

10x + 4y = 1220

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Translation: Roberto bought 6 cd's and 10 magazines for $900.00 pesos; In the same store, her friend María bought 10 CDs and 4

magazines at $1,220.00 pesos. What is the system of equations with two unknowns that represents the problem?

Find the indefinite integral. (Use c for the constant of integration.)

integral.gif

(2ti + j + 3k) dt

The indefinite **integral** of (2ti + j + 3k) dt is t^2i + tj + 3tk + C, where C is the constant of integration.

To find the **indefinite** integral of (2ti + j + 3k) dt, we integrate each component separately. The integral of 2ti with respect to t is (1/2)t^2i, as we increase the exponent by 1 and divide by the new exponent. The integral of j with respect to t is just tj, as j is a constant.

The integral of 3k with respect to t is 3tk, as k is also a **constant**. Finally, we add the constant of integration C to account for any potential constant terms. Therefore, the indefinite integral is t^2i + tj + 3tk + C.

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A receptionist can type documents 3 times as fast as her assistant. Working together, they can type up a day's worth of documents in 5 hours. On a day that the assistant is absent from work, find the number of hours, n, that it will take the receptionist to type up the day's documents on her own.

it will take the receptionist **20 hours** to type up the day's documents on her own when the assistant is absent.

When working together, their combined typing speed is (x + 3x) documents per hour, which is equal to 4x documents per hour.

Given that they can type up a day's worth of documents in 5 hours when working together, we can set up the following **equation**:

5 * 4x = 1

Simplifying the equation:

20x = 1

To find the receptionist's typing speed when working alone, we substitute x with 3x:

20 * 3x = 1

Simplifying the equation again:

60x = 1

Dividing both sides of the equation by 60:

x = 1/60

Therefore, the assistant's **typing speed** is 1/60 documents per hour.

To find the number of hours, n, it will take the receptionist to type up the day's documents on her own

n = 1 / (3x)

Substituting x with 1/60:

n = 1 / (3 * 1/60)

n = 1 / (1/20)

n = 20

Hence, it will take the receptionist **20 hours** to type up the day's documents on her own when the assistant is absent.

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Find the radius of convergence, R, of the series. [infinity] (x − 8)n n8 + 1 n = 0 .Find the interval of convergence, I, of the series. (Enter your answer using interval notation.)

The series converges on the **interval** from 7 inclusive to 9 exclusive.

To find the** radius** of convergence, we use the ratio test:

So the** **series converges absolutely if |x - 8| < 1, and diverges if |x - 8| > 1. Therefore, the radius of **convergence** is R = 1.

To find the interval of convergence, we need to test the endpoints x = 7 and x = 9:

When x = 7, the series becomes:

[infinity] (-1)ⁿ (n+9) / (n+1)

n = 0

which is an alternating series that satisfies the conditions of the alternating series test. Therefore, it converges.

When x = 9, the series becomes:

[infinity] 1 / (n+1)

n = 0

which is a** p-series** with p = 1, which diverges.

Therefore, the interval of convergence is [7, 9).

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Integrate the function ((x^2+y^2)^{frac{1}{3}}) over the region E that is bounded by the xy plane below and above by the paraboloid 10−7x^2−7y^2 using cylindrical coordinates.

∫∫∫E(x2+y2)13dV=∫BA∫DC∫FEG(z,r,θ) dzdrdθ∫∫∫E(x2+y2)13dV=∫AB∫CD∫EFG(z,r,θ) dzdrdθ

where A= , B= , C= , D= ,E= , F= and G(z,r,θ)= .The value of the integral is ∫∫∫E(x2+y2)13dV=∫

∫∫∫E(x^2+y^2)^(1/3) **dV** = ∫∫∫E(r^2)^(1/3) r dr dθ

**What is the integral of r^2^(1/3) over region E in cylindrical coordinates?**

In cylindrical coordinates, the given function ((x^2+y^2)^(1/3)) simplifies to (r^2)^(1/3) or r^(2/3). To integrate this function over the region E bounded by the xy plane and the paraboloid 10−7x^2−7y^2, we convert the **Cartesian** coordinates to cylindrical coordinates.

Let's rewrite the bounds in terms of cylindrical coordinates:

A = (0, 0, 0)

B = (r, θ, 0) (r > 0, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 10 - 7r^2)

C = (r, θ, z) (r > 0, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 10 - 7r^2)

D = (0, θ, 0) (0 ≤ θ ≤ 2π)

E = (r, θ, 0) (r > 0, 0 ≤ θ ≤ 2π)

F = (r, θ, 10 - 7r^2) (r > 0, 0 ≤ θ ≤ 2π)

G(z, r, θ) = r^(2/3)

Now, we can set up the triple **integral**:

∫∫∫E(r^2)^(1/3) r dr dθ = ∫₀²π ∫₀²√(10-z/7) r^(2/3) dr dθ ∫₀¹⁰-7r² dz

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Lauren dived a pild of paper into 5 stacks. The line plot shows the height of each stack of paper. What was the height, in inches, of the original paper?

A- 5/8 inches

B- 1 3/8 inches

C- 1 5/8 inches

D- 3 1/8 inches

The **height**, in inches, of the original paper include the following: C- 1 5/8 **inches**.

In Mathematics and Statistics, a **line plot** is a type of graph that is used for the graphical representation of data set above a number line, while using crosses, dots, or any other mathematical symbol.

Based on the information provided about the pile of** paper** that Lauren divided into 5 stacks, we would determine the** height** of the original **paper** in inches as follows;

Height of original paper = 1 + 1/8 + 4/8

Height of original paper = 1 + 5/8

**Height** of original **paper** = 1 5/8 **inches**.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

A parallelogram has sides 17. 3 m and 43. 4 m long. The height corresponding to the 17. 3-m base is 8. 7 m. Find the height, to the nearest tenth of a meter, corresponding to the 43. 4-m base

the **height** is 3.5m nearest tenth of a meter, corresponding to the 3.4-m base.

We know that the area of a **parallelogram** is given by A = base x height. Since the given parallelogram has two bases with different lengths, we will need to find the length of the other height to be able to calculate the area of the parallelogram.

Using the given measurements, let's call the 17.3m base as "b1" and its corresponding height as "h1", and call the 43.4m base as "b2" and its corresponding height as "h2".

From the given problem, we are given:

b1 = 17.3mh1 = 8.7m andb2 = 43.4m

Now, let's solve for h2:

Since the area of the parallelogram is the same regardless of which base we use, we can say that

A = b1*h1 = b2*h2 Substituting the given values, we have:

17.3m x 8.7m = 43.4m x h2

Simplifying: 150.51 sq m = 43.4m x h2h2 = 150.51 sq m / 43.4mh2 = 3.46636...

The height corresponding to the 43.4m base is 3.5m (rounded to the nearest tenth of a meter).Therefore, the height corresponding to the 43.4-m base is 3.5 meters.

Here, we are given that the parallelogram has sides of 17.3m and 43.4m, and its **corresponding height** is 8.7m. We are asked to find the length of the height corresponding to the 43.4m base.

Since the area of a parallelogram is given by A = base x height, we can use this formula to solve for the length of the other height of the parallelogram. We can call the 17.3m **base** as "b1" and its corresponding height as "h1", and call the 43.4m base as "b2" and its corresponding height as "h2".

Using the formula A = b1*h1 = b2*h2, we can find h2 by substituting the values we have been given.

Solving for h2, we get 3.46636.

Rounding to the nearest tenth of a meter, we get that the **length** of the height corresponding to the 43.4m base is 3.5m. Therefore, the answer is 3.5m.

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18 points here someone help me please

The **average atomic mass** of the element in the data table is given as follows:

28.1 amu.

How to calculate the mean of a data-set?The **mean **of a data-set is given by the sum of all observations in the data-set divided by the cardinality of the data-set, which represents the number of observations in the data-set.

For the **weighed mean**, we calculate the mean as the sum of each observation multiplied by it's weight.

Hence the **average atomic mass** of the element in the data table is given as follows:

0.922297 x 27.977 + 0.046832 x 28.976 + 0.030872 x 29.974 = 28.1 amu.

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381 . derive cosh2(x) sinh2(x)=cosh(2x) from the definition.

In order to derive cosh^2(x) sinh^2(x) = cosh(2x), we can use the definitions of **hyperbolic cosine and sine functions:**

cosh(x) = (e^x + e^(-x)) / 2

sinh(x) = (e^x - e^(-x)) / 2

We want to derive the identity cosh^2(x) sinh^2(x) = cosh(2x) using the **hyperbolic cosine and sine** definitions. First, we'll square the definitions of cosh and sinh:

cosh^2(x) = (e^x + e^(-x))^2 / 4

sinh^2(x) = (e^x - e^(-x))^2 / 4

Multiplying these **expressions** together, we get:

cosh^2(x) sinh^2(x) = (e^x + e^(-x))^2 / 4 * (e^x - e^(-x))^2 / 4

= (e^2x + 2 + e^(-2x)) / 16 * (e^2x - 2 + e^(-2x)) / 16

= (e^4x - 4 + 6 + e^(-4x)) / 256

= (e^4x + 2e^(-4x) + 2) / 16

Next, we'll use the identity cosh(2x) = cosh^2(x) + sinh^2(x) to express cosh(2x) in terms of cosh(x) and sinh(x):

cosh(2x) = cosh^2(x) + sinh^2(x)

= (e^x + e^(-x))^2 / 4 + (e^x - e^(-x))^2 / 4

= (e^2x + 2 + e^(-2x)) / 4

Now we can substitute this expression into our previous result:

cosh^2(x) sinh^2(x) = (e^4x + 2e^(-4x) + 2) / 16

= (cosh(2x) + 1) / 8

Thus we have shown that cosh^2(x) sinh^2(x) = (cosh(2x) + 1) / 8, which is equivalent to the identity cosh2(x) sinh2(x) = cosh(2x).

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Find largest part when £40is shared in the ratio5:3

**Answer:**

£25

**Step-by-step explanation:**

ratio 5:3 means there are 5 + 3 = 8 parts.

40/8 = 5.

we have 5(5) + 3(5) = 25 + 15 = 40.

the largest part is £25.

The figure below shows a rectangular window.

68 in

36 in

**Answer: If you need to find the area of the window it would be, 2,448.**

**Step-by-step explanation: To find the area you must multiply width times length.**

3. Let S= {a, b, c, d} be the sample space for an experiment. 3.1.Suppose the {a} is in the Sigma Algebra for the sample space. Is {b} necessarily in the Sigma Algebra? 3.2 .Suppose {a} and {b} are in the Sigma Algebra. Is the {c} necessarily in the Sigma Algebra?

3.1. No, {b} is not necessarily in the **Sigma Algebra** if {a} is.

3.2. No, {c} is not necessarily in the Sigma Algebra if {a} and {b} are.

Is {b} guaranteed to be in the Sigma Algebra if {a} is, and is {c} guaranteed to be in the Sigma Algebra if {a} and {b} are?In the **context** of the sample space S = {a, b, c, d} and the Sigma Algebra, we cannot conclude that {b} is necessarily in the Sigma Algebra if {a} is. Similarly, we cannot conclude that {c} is necessarily in the Sigma Algebra if both {a} and {b} are.

A Sigma Algebra, also known as a sigma-field or a Borel field, is a collection of subsets of the sample space that satisfies certain properties. It must contain the sample space itself, be closed under **complementation **(if A is in the Sigma Algebra, its complement must also be in the Sigma Algebra), and be closed under countable unions (if A1, A2, A3, ... are in the Sigma Algebra, their union must also be in the Sigma Algebra).

In 3.1, if {a} is in the Sigma Algebra, it means that the set {a} and its complement are both in the Sigma Algebra. However, this does not guarantee that {b} is in the Sigma Algebra because {b} may or may not satisfy the **properties **required for a set to be in the Sigma Algebra.

Similarly, in 3.2, even if {a} and {b} are both in the Sigma Algebra, it does not necessarily imply that {c} is also in the Sigma Algebra. Each set must individually satisfy the properties of the Sigma Algebra, and the presence of {a} and {b} alone does not determine whether {c} meets those requirements.

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Use algebra to rewrite the integrand; then integrate and simplify. (Use C for the constant of integration.) integral (3x^2 - 4)^2 x^3 dx Use algebra to rewrite the integrand; then integrate and simplify. (Use C for the constant of integration.) integral 3x + 3/x^7 dx

(a) After integrating and **simplification**, the ∫(3x² - 4)² x³ dx is 9(x⁸/8) - 24(x⁵/5) + 16(x⁴/4) + C, and also

(b) The **integral **∫(x + 3)/x⁷ dx is = (-1/5x⁵) - (1/2x⁶) + C.

Part(a) : We have to **integrate **: ∫(3x² - 4)² x³ dx,

We simplify using the **algebraic**-identity,

= ∫(9x² - 24x + 16) x³ dx,

= ∫9x⁷ - 24x⁴ + 16x³ dx,

On integrating,

We get,

= 9(x⁸/8) - 24(x⁵/5) + 16(x⁴/4) + C,

Part (b) : We have to **integrate **: ∫(x + 3)/x⁷ dx,

On simplification,

We get,

= ∫(x/x⁷ + 3/x⁷)dx,

= ∫(1/x⁶ + 3/x⁷)dx,

= ∫(x⁻⁶ + 3x⁻⁷)dx,

On integrating,

We get,

= (-1/5x⁵) - (3/6x⁶) + C,

= (-1/5x⁵) - (1/2x⁶) + C,

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The given question is incomplete, the complete question is

(a) Use algebra to rewrite the integrand; then integrate and simplify. (Use C for the constant of integration.)

∫(3x² - 4)² x³ dx,

(b) Use algebra to rewrite the integrand; then integrate and simplify. (Use C for the constant of integration.)

∫(x + 3)/x⁷ dx.

s it appropriate to use a regression line to predict y-values for x-values that are not in (or close to) the range of x-values found in the data?

A. It is appropriate because the regression line models a trend, not the actual points, so although the prediction of the y-value may not be exact it will be precise. B. It is appropriate because the regression line will always be continuous, so a y value exists for every x-value on the axis. C. It is not appropriate because the correlation coefficient of the regression line may not be significant. D. It is not appropriate because the regression line models the trend of the given data, and it is not known if the trend continues beyond the range of those data.

It is important to consider the limitations of the **regression** **line** and the potential consequences of extrapolation before making any predictions outside of the range of observed data. Option D is the correct answer.

The answer to whether it is appropriate to use a regression line to predict y-values for x-values that are not in (or close to) the **range** of x-values found in the data depends on the context and purpose of the analysis.

However, in general, option D, "It is not appropriate because the regression line models the trend of the given data, and it is not known if the trend **continues** beyond the range of those data" is the most accurate.

The regression line represents the trend observed in the given data and is not necessarily indicative of what may happen outside of that range.

Extrapolating beyond the range of data can lead to unreliable **predictions**, and it is better to use caution and only make predictions within the range of observed data.

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D. It is not appropriate because the **regression line** models the trend of the given data, and it is not known if the trend continues beyond the range of those** data.**

D. It is not appropriate because the r**egression line** models the trend of the given data, and it is not known if the trend continues beyond the range of those data. The **regression line** is based on the values within the range of the data, and extrapolating outside of that range may not accurately reflect the trend. It is important to consider the limitations of the data and the model when using regression to make predictions.

The term "regression" was coined by Francis Galton in the 19th century to describe a biological phenomenon. The result is that the **height** of descendants of higher ancestors returns to the original **mean **(this phenomenon is also called regression to the mean).

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The arrival rate for a certain waiting-line system obeys a Poisson distribution with a mean of 0.5 unit per period. It is required that the probability of one or more units in the system not exceed 0.20. What is the minimum service rate that must be provided if the service duration is to be distributed exponentially?

The **minimum service rate** that must be provided is 1.609 units per period.

To solve this problem, we need to use the **M/M/1 queueing model**, where the arrival process follows a** Poisson distribution**, the service process follows an **exponential distribution**, and there is one server.

We can use **Little's law** to relate the average number of units in the system to the arrival rate and the average service time:

**L = λ * W**

where L is the **average number of units** in the system, λ is the **arrival rate**, and W is the **average time spent** in the system.

From the problem statement, we want to find the **minimum service rate **in the system not exceeding 0.20. This means that we want to find the **maximum value** of W such that P(W ≥ 0.20) ≤ 0.80.

Using the** M/M/1 queueing model**, we know that the average time spent in the system is:

**W = Wq + 1/μ**

where Wq is the average time spent waiting in the queue and μ is the service rate.

Since we want to find the minimum service rate, we can assume that there is **no waiting** in the queue (i.e., Wq = 0).

Plugging in Wq = 0 and λ = 0.5 into** Little's law**, we get:

**L = λ * W = λ * (1/μ)**

Since we want P(W ≥ 0.20) ≤ 0.80, we can use the **complementary probability:**

P(W < 0.20) ≥ 0.20

Using the formula for the **exponential distribution**, we can calculate:

P(W < 0.20) = 1 - e^(-μ * 0.20)

Setting this expression greater than or equal to 0.20 and solving for μ, we get:

**μ ≥ -ln(0.80) / 0.20 ≈** **1.609**

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Study these equations: f(x) = 2x – 4 g(x) = 3x 1 What is h(x) = f(x)g(x)? h(x) = 6x2 – 10x – 4 h(x) = 6x2 – 12x – 4 h(x) = 6x2 2x – 4 h(x) = 6x2 14x 4.

The correct answer is "h(x) = 6x² - 12x." The other options you listed do not match the correct **expression **obtained by** multiplying** f(x) and g(x).

To find h(x) = f(x)g(x), we need to** multiply** the equations for f(x) and g(x):

f(x) = 2x - 4

g(x) = 3x

Multiplying these equations gives:

h(x) = f(x)g(x) = (2x - 4)(3x)

Using the **distributive property**, we can expand this expression:

h(x) = 2x × 3x - 4 × 3x

h(x) = 6x² - 12x

So, the correct expression for h(x) is h(x) = 6x² - 12x.

Among the options you provided, the correct answer is "h(x) = 6x² - 12x." The other options you listed do not match the correct **expression **obtained by multiplying f(x) and g(x).

It's important to note that the equation h(x) = 6x² - 12x represents a **quadratic function**, where the highest power of x is 2. The coefficient 6 represents the quadratic term, while the coefficient -12 represents the linear term.

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katie wants to cover this prism in glitter if 60 of glitter is needed to cover each m square how much glitter will she need to cover the prism completely

The amount of **glitter** that is needed to **cover** the prism completely is 87.6 kg.

In Mathematics, the** surface area** of a** triangular prism** can be calculated by using this mathematical expression:

Total surface area of triangular prism = (Perimeter of the base × Length of the prism) + (2 × Base area)

Total surface area of **triangular prism** = (S₁ + S₂ + S₃)L + bh

where:

b represent the bottom edge of the base triangle.h is the height of the base triangle.L represent the length of the triangular prism.S₁, S₂, and S₃ represent the three sides (edges) of the base triangle.By substituting the given side lengths into the formula for the **surface area** of a** triangular prism**, we have the following;

Total surface area of **triangular prism** = (13 × 25) + (1/2 × 21 × 10 × 2) + (16 × 25) + (21 × 25)

Total surface area of **triangular prism** = 325 + 210 + 400 + 525

Total surface area of **triangular prism** = 1,460 m².

For the amount of **glitter** that is needed, we have:

Amount of **glitter** = (60 × 1,460)/1000

Amount of **glitter** = 87.6 kg.

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test the given set of solutions for linear independence. differential equation solutions y'' y = 0 {sin(x), sin(x) − cos(x)} linearly independent linearly dependent

The solutions {sin(x), sin(x) - cos(x)} are** linearly Independent **since the linear combination equals zero only when all the coefficients are zero

To test the given set of solutions {sin(x), sin(x) - cos(x)} for linear independence, we can check if the linear combination of the solutions equals the zero **vector **only when all the coefficients are zero.

Let's consider the linear combination:

c1sin(x) + c2(sin(x) - cos(x)) = 0

Expanding this equation:

c1sin(x) + c2sin(x) - c2*cos(x) = 0

Rearranging terms:

sin(x)*(c1 + c2) - cos(x)*c2 = 0

This equation holds for all x if and only if both the coefficients of sin(x) and cos(x) are zero.

From the **equation**, we have:

c1 + c2 = 0

-c2 = 0

Solving this system of equations, we find that c1 = 0 and c2 = 0. This means that the only solution to the linear combination is the trivial solution, where all the **coefficients **are zero

Therefore, the solutions {sin(x), sin(x) - cos(x)} are linearly independent since the linear combination equals zero only when all the coefficients are zero

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The only solution to the **linear combination** being equal to zero is when both coefficients are zero. Hence, the given set of solutions {sin(x), sin(x) − cos(x)} is linearly **independent**.

To test the given set of solutions for linear independence, we need to check whether the linear combination of these solutions equals zero only when all coefficients are **zero**.

Let's write the linear combination of the given solutions:

c1 sin(x) + c2 (sin(x) - cos(x))

We need to find** whether **there exist non-zero coefficients c1 and c2 such that this linear combination equals zero for all x.

If we simplify this expression, we get:

(c1 + c2) sin(x) - c2 cos(x) = 0

For this equation to hold for all x, we must have:

c1 + c2 = 0 and c2 = 0

The second equation implies that c2 must be zero. Substituting this into the first equation, we get:

c1 = 0

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evaluate the following expression over the interval [−π2,π2]. arcsin(−3‾√2)

To evaluate the **expression** arcsin(-3√2) over the interval [-π/2,π/2], we need to find the angle θ that satisfies sin(θ) = -3√2.

Since sin is negative in the second and **third quadrants**, we can narrow down the possible values of θ to the interval [-π, -π/2) and (π/2, π].

To find the exact value of θ, we can use the **inverse sine** function, also known as arcsine:

θ = arcsin(-3√2) = -1.177 radians (rounded to three decimal places)

Since -π/2 < θ < π/2, the angle θ is within the given interval [-π/2, π/2].

Therefore, the evaluated expression is -1.177 radians.

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Let X be a continuous random variable with PDF:fx(x) = 4x^3 0 <= x <=10 otherwiseIf Y = 1/X, find the PDF of Y.If Y = 1/X, find the PDF of Y.

We know that the **probability density function** of Y is:

f y(y) =

{-4/y^5 y > 0

{0 otherwise

To find the probability density function (PDF) of Y, we need to first find the **cumulative distribution function** (CDF) of Y and then differentiate it with respect to Y.

Let Y = 1/X. Solving for X, we get X = 1/Y.

Using the change of variables method, we have:

Fy(y) = P(Y <= y) = P(1/X <= y) = P(X >= 1/y) = 1 - P(X < 1/y)

Since the PDF of X is given by:

fx(x) =

{4x^3 0 <= x <=10

{0 otherwise

We have:

P(X < 1/y) = ∫[0,1/y] 4x^3 dx = [x^4]0^1/y = (1/y^4)

Therefore,

Fy(y) = 1 - (1/y^4) = (y^-4) for y > 0.

To find the PDF of Y, we **differentiate **the CDF with respect to Y:

f y(y) = d(F) y(y)/d y = -4y^-5 = (-4/y^5) for y > 0.

Therefore, the PDF of Y is:

f y(y) =

{-4/y^5 y > 0

{0 otherwise

This is the final answer.

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use implicit differentiation to find an equation of the tangent line to the curve at the given point

sin(x+y) = 2x-2y (pi,pi)

x^2 + 2xy -y^2 +x= 2 (1,2) hyperbola

Using **implicit differentiation, **The equation of the tangent line to the curve at (1, 2) is: y = (-1/3)x + (7/3)

For the **curve** sin(x+y) = 2x-2y at the point (pi, pi):

Taking the derivative of both sides with respect to x using the chain rule, we get:

cos(x+y) (1 + dy/dx) = 2 - 2dy/dx

Simplifying, we get:

dy/dx = (2 - cos(x+y)) / (2 + cos(x+y))

At the point (pi, pi), we have x = pi and y = pi, so cos(x+y) = cos(2pi) = 1.

Therefore, the **slope** of the tangent line at (pi, pi) is:

dy/dx = (2 - cos(x+y)) / (2 + cos(x+y)) = (2 - 1) / (2 + 1) = 1/3

Using the **point-slope** form of the equation of a line, the equation of the tangent line at (pi, pi) is:

y - pi = (1/3)(x - pi)

Simplifying, we get:

y = (1/3)x + (2/3)pi

For the **hyperbola** x^2 + 2xy - y^2 + x = 2 at the point (1, 2):

Taking the **derivative** of both sides with respect to x using the product rule, we get:

2x + 2y + 2xdy/dx + 1 = 0

Solving for dy/dx, we get:

dy/dx = (-x - y - 1) / (2x + 2y)

At the point (1, 2), we have x = 1 and y = 2, so the **slope** of the tangent line at (1, 2) is:

dy/dx = (-x - y - 1) / (2x + 2y) = (-1-2-1)/(2+4) = -2/6 = -1/3

Using the **point-slope** form of the equation of a line, the equation of the tangent line at (1, 2) is:

y - 2 = (-1/3)(x - 1)

Simplifying, we get:

y = (-1/3)x + (7/3)

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Bonnie deposits $70. 00 into a saving account and the account earns 4. 5% simple interest a year no money is added or taken out for 3 years how much money does Bonnie have at the end of 3 years?

We can calculate the amount of money Bonnie will have in her savings account after 3 years using the** simple interest** formula:

A = P(1 + rt)

where A is the total amount of money at the end of the time period, P is the initial principal or deposit, r is the annual interest rate (as a decimal), and t is the time period in years.

In this case, Bonnie deposits $70.00 into her savings account and earns 4.5% simple interest a year. We know that she does not add or take out any money for 3 years. Therefore:

P = $70.00

r = 0.045 (since the interest rate is given as a percentage, we need to divide by 100 to get the decimal form)

t = 3 years

Plugging these values into the formula, we get:

A = $70.00(1 + 0.045 x 3)

A = $70.00(1.135)

A = $79.45

Therefore, Bonnie will have $79.45 in her savings account at the end of 3 years.

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Plot this into a graph.

y = tan (x + 90°) - 1

The **graph **is **attached **below.

To plot the **graph **of the equation y = tan(x + 90°) - 1, we can follow these steps:

x | y = tan(x + 90°) - 1

-180° | undefined

-135° | 1

-90° | 0

-45° | -1

0° | undefined

45° | -1

90° | 0

135° | 1

180° | undefined

Note: The values are given in degrees.

Plot theHere is a **graph** of the equation y = tan(x + 90°) - 1 is attached.

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Mateo is filling a cylinder-shaped swimming pool that has a diameter of

20 feet and a height of 4. 5 feet. He fills it with water to a depth of 3 feet.

The **volume **of water in the pool is 942 cubic feet.

Here, we have

Given:

A swimming pool with a **diameter **of 20 feet and a height of 4.5 feet is being filled by Mateo. He adds water till it is 3 feet deep. The pool's water volume must be determined.

Use the formula for the volume of a **cylinder**, which is provided as V = r2h, to get the volume of the cylinder pool. V stands for the cylinder's volume, r for its **radius**, h for its height, and for pi number, which is 3.14.

Here, we have a diameter = 20 feet.

As a result, the cylinder's radius is equal to 10 feet, or half of its diameter.

We are also informed that the cylinder has a **height **of 4.5 feet and a depth of 3 feet.

As a result, the pool's water level is 3 feet high. When the values are substituted into the **formula**, we get:

V = πr²h = 3.14 x 10² x 3 = 942 cubic feet

Therefore, the volume of water in the pool is 942 cubic feet.

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Find the Fourier series of the given function f(x), which is assumed to have the period 21. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.

1. f(x) = x2 = (-1 < x < TT)

The** Fourier series** for f(x) is: f(x) = \frac{\pi^2}{3} + \sum_{n=1}^{\infty} \frac{2}{n^2} \cos(nx)$

The Fourier series of f(x) = x^2, where -π < x < π, can be** found** using the formula:

$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 dx = \frac{\pi^2}{3}$

$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) dx = \frac{2}{n^2}$

$b_n = 0$ for all n, since f(x) is an even** function**

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The first three terms of a sequence are given. Round to the nearest thousandth (if necessary). 6, 9,12

To find the pattern in the given **sequence**, we can observe that each term increases by 3.

Using this pattern, we can **determine** the next terms of the sequence:

6, 9, 12, 15, 18, ...

So the first three terms are 6, 9, and 12.Starting with the first term, which is 6, we add 3 to get the second term: 6 + 3 = 9.

Similarly, we add 3 to the** second term t**o get the third term: 9 + 3 = 12.

If we continue this pattern, we can find the next terms of the sequence by adding 3 to the **previous term**:

12 + 3 = 15

15 + 3 = 18

18 + 3 = 21

...

So, the sequence continues with 15, 18, 21, and so on, with each term obtained by adding 3 to the previous term.

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find the mean, median m and mode of the word problem show your work and write answers in space provided the school nurse recorded the height in inches of eight grade 5 numbers 50, 51 , 56 ,52 , 57,60,62

**Answer:**

mean=sum of all numbers /total no of data

50+51+56+52+57+60+62/7

388/7

55.42857. or 55 3/7

tom's major predicament is one of the themes of the play. this theme could best be described as
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