He pays $4 for parking and $12 for each pizza he buys. If he plays a total of $52 how many pizzas did he buy

The weekly CPU time used by the firm is described by a probability density function, and we can use this function to find the expected value and variance of the CPU time used. Furthermore, we can use these values to find the expected value and variance of the weekly cost for CPU time.

Expected Value and Variance are statistical measures that help us understand the central tendency and variability of a random variable, respectively. The expected value of a random variable is its average value, while the variance is a measure of how spread out the values are around the mean.

A) To find the expected value and variance of the CPU time used, we can use the following formulas:

Expected Value (E(Y)) = ∫ y*f(y) dy, where f(y) is the probability density function

Variance (V(Y)) = E(Y²) - (E(Y))²

For the given probability density function,

f(y) = {(3/64)y²* (y-4) 0 ≤ y ≤ 4},

we can substitute this into the above formulas and integrate from 0 to 4 to get:

E(Y) = ∫ yf(y) dy = ∫ y(3/64)y² * (y-4) dy = 3/4

V(Y) = E(Y²) - (E(Y))² = ∫ y²*f(y) dy - (3/4)² = 3/16

Therefore, the expected value of CPU time used per week is 0.75 hours, and the variance is 0.1875 hours².

B) To find the expected value and variance of the weekly cost for CPU time, we can use the fact that the CPU time costs the firm $200 per hour. Thus, the cost of CPU time per week can be represented as [tex]Y_{c}[/tex] = 200*Y, where Y is the CPU time used per week. Therefore,

E([tex]Y_{c}[/tex]) = E(200Y) = 200E(Y) = $150

V([tex]Y_{c}[/tex]) = V(200*Y) = (200²)*V(Y) = $7500

Hence, the expected weekly cost for CPU time is $150, and the variance is $7500.

C) To determine whether the weekly cost would exceed $600 very often, we can use Chebyshev's inequality, which tells us that for any random variable, the probability that its value deviates from the expected value by more than k standard deviations is at most 1/k². In other words, the probability of an extreme event decreases rapidly as we move away from the mean.

Using this inequality, we can say that the probability of the weekly cost exceeding $600 by more than k standard deviations is at most 1/k². For example, if we want the probability to be at most 0.01 (1%), we can choose k = 10. Thus, the probability that the weekly cost exceeds $600 by more than 10 standard deviations is at most 1/10² = 0.01, or 1%.

Therefore, we can conclude that it is unlikely for the weekly cost to exceed $600 very often, given the probability density function and the expected value and variance of the weekly cost that we have calculated.

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Complete Question

Weekly CPU time used by an accounting firm has a probability density function (measured in hours) given by: f(y)={(3/64)y^2 * (y-4) 0 <= y <= 4={0 elsewhere

A) Find the E(Y) and V(Y)

B) The CPU time costs the firm $200 per hour. Find E(Y) and V(Y) of the weekly cost for CPU time.

C) Would you expect the weekly cost to exceed $600 very often? Why?

The area of the regular octagon inscribed in a circle with a radius of 16 feet can be found using the formula A = (2 + 2sqrt(2))r^2, where r is the radius of the circle. Plugging in the value for r, we get:

A = (2 + 2sqrt(2))(16)^2

A = (2 + 2sqrt(2))(256)

A = 660.254 ft^2

Therefore, the area of the regular octagon is approximately 660.254 square feet.

To derive the formula for the area of a regular octagon inscribed in a circle, we can divide the octagon into eight congruent isosceles triangles, each with a base of length r and two congruent angles of 22.5 degrees. The height of each triangle can be found using the sine function, which gives us h = r * sin(22.5). Since there are eight of these triangles, the area of the octagon can be found by multiplying the area of one of the triangles by 8, which gives us:

A = 8 * (1/2)bh

A = 8 * (1/2)(r)(r*sin(22.5))

A = 4r^2sin(22.5)

We can simplify this expression using the double angle formula for sine, which gives us:

A = 4r^2sin(45)/2

A = (2 + 2sqrt(2))r^2

Therefore, the formula for the area of a regular octagon inscribed in a circle is A = (2 + 2sqrt(2))r^2.

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The slope of a line n is 0. Therefore, option B is the correct answer.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

Slope of a horizontal line. When two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0, and you will also find this by using the slope formula.

Therefore, option B is the correct answer.

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For n=4, the possible values of ℓ (angular momentum quantum number) are 0, 1, 2, and 3. Therefore, the answer is 0, 1, 2, 3.
For n=4, the possible values of ℓ are determined by the equation ℓ = 0 to (n-1). To find the possible values of ℓ, follow these steps:

1. Start with ℓ = 0.
2. Increase ℓ by 1 until you reach (n-1).

For n=4, the values of ℓ are:

ℓ = 0, 1, 2, 3

These are the possible values of ℓ in ascending order, separated by commas.

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The length of the curve represented by the vector function r(t) = 6t, t^2, 19t^3, where 0 ≤ t ≤ 1, is approximately 27.9865. To find the length of the curve represented by the vector function r(t) = 6t, t^2, 19t^3, where 0 ≤ t ≤ 1, we need to use the formula for arc length of a vector function.

This formula is given by:

L = ∫a^b ||r'(t)|| dt

where L is the length of the curve, a and b are the lower and upper bounds of the parameter t, and ||r'(t)|| is the magnitude of the derivative of r(t) with respect to t.

In this case, we have:

r(t) = 6t, t^2, 19t^3
r'(t) = 6, 2t, 57t^2
||r'(t)|| = √(6^2 + (2t)^2 + (57t^2)^2)
||r'(t)|| = √(36 + 4t^2 + 3249t^4)

Now we can substitute these expressions into the formula for arc length and integrate:

L = ∫0^1 √(36 + 4t^2 + 3249t^4) dt

This integral is not easy to solve analytically, so we need to use numerical methods to approximate the answer. One common method is to use Simpson's rule, which gives:

L ≈ h/3 [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)]

where h is the step size (h = (b-a)/n), f(xi) is the value of the integrand at the ith interval endpoint, and n is the number of intervals (n must be even).

Using Simpson's rule with n = 100 (for example), we get:

L ≈ 27.9865

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The graph that represents [tex]y = \sqrt[3]{x - 5}[/tex] is given by the image presented at the end of the answer.

The parent function in the context of this problem is defined as follows:

[tex]y = \sqrt[3]{x}[/tex]

The translated function in the context of this problem is defined as follows:

[tex]y = \sqrt[3]{x - 5}[/tex]

The translation is defined as follows:

x -> x - 5, meaning that the function was translated five units right.

Hence the vertex of the function is moved from (0,0) to (5,0), as shown on the image given at the end of the answer.

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In triangle ABC, where angle A is 46°, facet b is 8, perspective B is 24°, angle C is 90°, and aspect c is 13, angle c is 110° and facet a is approximately 9.95

To discover angle C and facet A in triangle ABC, we are able to use the residences of triangles and trigonometric ratios. Given the following statistics:

Angle A = 46°

Side b = 8

Angle B = 24°

Angle C = 90° (Right Angle)

Side c = 13

Next, we can use the sine ratio to locate side a:

sin(A) = contrary / hypotenuse

sin(46°) = a / 13

Rearranging the equation to solve for facet a:

a = 13 * sin(46°)

a ≈ 9.95

Therefore, in triangle ABC, where angle A is 46°, facet b is 8, perspective B is 24°, angle C is 90°, and aspect c is 13, and facet A is approximately 9.95

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The correct question is;

A = 46°, b=8

B = 24°

"Find out aspect A, aspect C, and angle c in ABC"

The answer is:

(a) and (b) would produce categorical data.

(b) which are categorical responses.

(c) would produce quantitative data

(d) would also be answered with a numerical response, which is quantitative.

What is statistics?

Statistics is the branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It involves the use of mathematical methods to gather, summarize, and interpret data, which can be used to make decisions or draw conclusions about a population based on a sample of that population.

Categorical data refers to data that can be divided into categories or groups.

The categories are usually non-numerical, although they can be represented using numerical codes.

The categories are often based on qualitative characteristics or attributes, such as color, gender, or type of animal.

In the examples given:

(a) and (b) would produce categorical data.

(a) "What is your height?" could be answered with categorical options such as "short," "medium," or "tall."

(b) "Do you have any pets?" could be answered with a simple "yes" or "no," which are categorical responses.

(c) and (d) would produce quantitative data.

(c) "How many pets do you have?" would be answered with a numerical response, which is quantitative.

(d) "How many books did you read last year?" would also be answered with a numerical response, which is quantitative.

Hence, the answer is:

(a) and (b) would produce categorical data.

(b) which are categorical responses.

(c) would produce quantitative data

(d) would also be answered with a numerical response, which is quantitative.

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The present value of an annuity that pays $100 every six months for five years, with an interest rate of 8% per year compounded monthly, is approximately $1,901.22.

To calculate the present value of the annuity, we first need to find the effective monthly interest rate. This can be calculated by dividing the annual interest rate by 12 and then converting it to a decimal:

r = 8% / 12 = 0.00666666667

Next, we calculate the number of periods for the annuity:

n = 5 years x 2 periods per year = 10 periods

Using the formula for the present value of an annuity, we can calculate the present value of the annuity:

PV = payment x ((1 - (1 + r)^-n) / r)

Substituting the values we have calculated, we get:

PV = $100 x ((1 - (1 + 0.00666666667)^-10) / 0.00666666667) = $1,901.22

Therefore, the present value of the annuity is approximately $1,901.22.

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The area between Z= -2.68 and Z= -0.99 for the standard normal distribution is 0.3389. (option c)

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The area under the curve of the standard normal distribution represents the probability of a random variable taking a certain value or falling within a certain range.

To find the area between two values of the standard normal distribution, we can use a standard normal table or a calculator with a standard normal distribution function. In this case, we can use a standard normal table to find the area between Z= -2.68 and Z= -0.99.

The table gives us the area to the left of Z= -2.68 as 0.0038 and the area to the left of Z= -0.99 as 0.1611. To find the area between Z= -2.68 and Z= -0.99, we subtract the area to the left of Z= -2.68 from the area to the left of Z= -0.99:

0.1611 - 0.0038 = 0.1573

Therefore, the area between Z= -2.68 and Z= -0.99 for the standard normal distribution is approximately 0.1573 or 0.3389 when rounded to four decimal places.

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Using the formula for uniformly decelerated motion, the distance traveled until reaching rest can be calculated as 140.625 meters.

To calculate the distance traveled until the train comes to a stop, we can use the equation of motion for uniformly decelerated motion. The equation is:

v² = u² + 2as

Where:

v = final velocity (0 m/s, since the train comes to a stop)

u = initial velocity (25 m/s)

a = acceleration (negative, as it's decelerating uniformly)

s = distance traveled

Rearranging the equation, we get:

s = (v² - u²) / (2a)

Plugging in the values:

s = (0² - 25²) / (2a)

Since the train slows down uniformly, the acceleration can be calculated as the change in velocity divided by the distance:

a = (5 - 25) / 90

Plugging this back into the equation:

s = (0² - 25²) / (2 * ((5 - 25) / 90))

Simplifying further:

s = -625 / (-40 / 9) = 140.625 m

Therefore, the distance traveled until the train comes to a stop is approximately 140.625 meters.

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The annual interest rate charged was approximately 6.75%. To calculate the annual interest rate charged, we can use the simple interest formula:

I = P * r * t

where I is the interest charged, P is the principal amount, r is the annual interest rate, and t is the time period in years.

In this case, we know that the principal amount is $4000, the time period is 10/12 years (since the loan was repaid after 10 months), and the total amount repaid is $4270. To find the interest charged, we can subtract the principal amount from the total amount:

I = $4270 - $4000 = $270

Substituting these values into the simple interest formula, we get:

$270 = $4000 * r * (10/12)

Simplifying this equation, we get:

r = $270 / ($4000 * 10/12) = 0.0675 or 6.75%

Therefore, the annual interest rate charged was approximately 6.75%.

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The equation of the plane is 14x - 9y - 16z = -22.

To find the equation of a plane, we need a point on the plane and the normal vector to the plane. Since the plane passes through the point (8, 0, -3), we know that any point on the plane will satisfy the equation 14x - 9y - 16z = k for some constant k. We can use the coordinates of the point to find k: 14(8) - 9(0) - 16(-3) = 182. So the equation of the plane is 14x - 9y - 16z = 182.

Alternatively, we can find two points on the plane (by setting t = 0 and t = 1 in the equation of the line) and then use their cross product to find the normal vector to the plane. The two points are (5, 0, 4) and (2, 1/4, 4/3). Their cross product is (-9/4, -16, 45/4), which is a normal vector to the plane. Dividing by the GCD of the coefficients, we get the equation 14x - 9y - 16z = -22. So, the equation of the plane is 14x - 9y - 16z = -22.

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The general indefinite integral of `sec(t)(3 sec(t) 8 tan(t)) dt` is `3t + 3/2 tan^2(t) + 4 ln|sec(t) + tan(t)| + C`.

To find the indefinite integral of `sec(t)(3 sec(t) 8 tan(t)) dt`, we can use the distributive property of multiplication to expand the expression inside the parentheses, and then use the trigonometric identity `sec^2(t) = 1 + tan^2(t)` to simplify the integrand:

```

sec(t)(3 sec(t) 8 tan(t)) dt

= 3 sec^2(t) dt + 8 sec(t) tan(t) dt    (distribute sec(t))

= 3 (1 + tan^2(t)) dt + 8 sec(t) tan(t) dt    (use sec^2(t) = 1 + tan^2(t))

= 3 dt + 3 tan^2(t) dt + 8 sec(t) tan(t) dt    (expand)

```

Now we can integrate each term separately:

```

∫ sec(t)(3 sec(t) 8 tan(t)) dt

= ∫ 3 dt + ∫ 3 tan^2(t) dt + ∫ 8 sec(t) tan(t) dt

= 3t + 3/2 tan^2(t) + 4 ln|sec(t) + tan(t)| + C   (where C is the constant of integration)

```

Therefore, the general indefinite integral of `sec(t)(3 sec(t) 8 tan(t)) dt` is `3t + 3/2 tan^2(t) + 4 ln|sec(t) + tan(t)| + C`.

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The function that models the population of foxes in the Arctic circle at time t (in months) since the beginning of Alyssa's study is P(t) = 185,000 * (17/18)^(t/2).

To model the population of foxes in the Arctic circle over time, we can use exponential decay since the population loses 1/18 (start fraction, 1, divided by, 18, end fraction) of its size every 2/22 months.

Let P(t) represent the population of foxes at time t (in months) since the beginning of Alyssa's study. The initial population is given as 185,000 (185,000185, comma, 000 foxes).

The exponential decay function can be written as:

P(t) = P₀ * (1 - r)^n

Where:

P₀ is the initial population (185,000 in this case).

r is the decay rate per time period (1/18 in this case).

n is the number of time periods elapsed (t/2).

Plugging in the values, the function that models the population of foxes over time becomes:

P(t) = 185,000 * (1 - 1/18)^(t/2)

Simplifying further:

P(t) = 185,000 * (17/18)^(t/2).

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The area function Ap(x) represents the area of the region under the curve y = t³ between the vertical lines y = p and t = x. To find the formula for Ap(x), we need to integrate the function y = t³ with respect to t between the limits p and x.

∫[p,x] t³ dt = [t⁴/4]pᵡ

Now, substitute x for t in the above expression and subtract the result obtained by substituting p for t.

Ap(x) = [(x⁴/4) - (p⁴/4)]

Therefore, the formula for the area function Ap(x) is Ap(x) = (x⁴/4) - (p⁴/4). This formula depends on x and the parameter p, which represents the vertical line y = p.

In simpler terms, Ap(x) is the area of the shaded region between the curve y = t³ and the vertical lines y = p and t = x. The formula for Ap(x) is obtained by integrating the function y = t³ with respect to t and subtracting the result obtained by substituting p for t from the result obtained by substituting x for t.

In summary, the area function Ap(x) represents the area of the region under the curve y = t³ between the vertical lines y = p and t = x. The formula for Ap(x) is (x⁴/4) - (p⁴/4), which depends on x and the parameter p.

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This is the indefinite integral of x^3√(x^2 - 36) dx using the substitution x = 6secθ, with C representing the constant of integration.

To solve the indefinite integral ∫x^3√(x^2 - 36) dx using the substitution x = 6secθ, we can follow these steps:

Step 1: Find the derivative of x = 6secθ with respect to θ.

dx/dθ = 6secθtanθ

Step 2: Rearrange the substitution equation to solve for dx.

dx = 6secθtanθ dθ

Step 3: Substitute x and dx in terms of θ into the original integral.

∫(6secθ)^3 √((6secθ)^2 - 36) (6secθtanθ) dθ

Step 4: Simplify the expression.

∫216sec^3θ √(36sec^2θ - 36) tanθ dθ

Step 5: Use trigonometric identities to simplify further.

Recall that sec^2θ - 1 = tan^2θ.

Therefore, 36sec^2θ - 36 = 36tan^2θ.

∫216sec^3θ √(36tan^2θ) tanθ dθ

= ∫216sec^3θ |6tanθ| tanθ dθ

= 1296 ∫sec^3θ |tan^2θ| dθ

Step 6: Evaluate the integral using the power rule for integrals.

Recall that ∫sec^3θ dθ = (1/2)(secθtanθ + ln|secθ + tanθ|) + C.

Therefore, we have:

= 1296 [(1/2)(secθtanθ + ln|secθ + tanθ|) - (1/2)ln|cosθ|] + C

Step 7: Convert back to the original variable x.

Recall that x = 6secθ, and we can use the Pythagorean identity sec^2θ = 1 + tan^2θ to simplify the expression.

= 1296 [(1/2)(x + ln|x + √(x^2 - 36)|) - (1/2)ln|√(x^2 - 36)/6|] + C

Simplifying further:

= 648(x + ln|x + √(x^2 - 36)| - ln|√(x^2 - 36)/6|) + C

This is the indefinite integral of x^3√(x^2 - 36) dx using the substitution x = 6secθ, with C representing the constant of integration.

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a. the probability of p of not entering the store before the first customer. b. the probability that the fourth person will be the first customer to enter the store is 0.093. c. the probability that it will take more than three people to pass by before the first customer enters the store is 0.328

(a) We are given that the probability of a person entering a store with a creative window display is 0.61. Let X be the number of people who walk by before the first customer enters the store. Then X follows a geometric distribution with parameter p = 0.61, since each person has a probability of p of not entering the store before the first customer.

(b) The probability that the fourth person will be the first customer to enter the store is given by P(X=3), since X represents the number of people who pass by before the first customer enters the store. Using the formula for the geometric distribution, we have:

P(X=3) = (1-p)^(3-1) * p = (0.39)^2 * 0.61 = 0.093

Therefore, the probability that the fourth person will be the first customer to enter the store is 0.093.

(c) The probability that it will take more than three people to pass by before the first customer enters the store is given by P(X>3). Using the formula for the geometric distribution, we have:

P(X>3) = 1 - P(X<=3) = 1 - [P(X=1) + P(X=2) + P(X=3)]

= 1 - [p + (1-p)p + (1-p)^2p]

= 1 - [0.61 + 0.390.61 + 0.39^20.61]

= 1 - 0.67177

= 0.328

Therefore, the probability that it will take more than three people to pass by before the first customer enters the store is 0.328, rounded to three decimal places.

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Answer:

32.38°

Step-by-step explanation:

Naoya has a 70% probability of succeeding at any given free-throw. In a sample of 15 free-throws, he can expect to succeed in 10.5 free-throws on average.

The probability of Naoya succeeding at any given free-throw is 0.7, or 70%. To find the expected number of free-throws he can succeed in a sample of 15, we use the formula for the expected value of a binomial distribution.

The number of trials is 15, the probability of success is 0.7, and we want to find the expected number of successes. The formula for the expected value of a binomial distribution is E(X) = n*p, where E(X) is the expected number of successes, n is the number of trials, and p is the probability of success.

E(X) = 15*0.7 = 10.5.

Therefore, Naoya can expect to succeed in 10.5 free-throws on average in a sample of 15 free-throws.

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There is a 70% probability of succeeding at any given free-throw. In a sample of 15 free-throws, he can expect to succeed in 10.5 free-throws on average.

The probability of Naoya succeeding at any given free-throw is 0.7, or 70%. To find the expected number of free-throws he can succeed in a sample of 15, we use the formula for the expected value of a binomial distribution.

The number of trials is 15, the probability of success is 0.7, and we want to find the expected number of successes. The formula for the expected value of a binomial distribution is E(X) = n*p, where E(X) is the expected number of successes, n is the number of trials, and p is the probability of success.

E(X) = 15*0.7 = 10.5.

Therefore, Naoya can expect to succeed in 10.5 free-throws on average in a sample of 15 free-throws.

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The resultant answer after solving the function is:
  a^(-1)(t) = [t, 0, t^(1/5), t/2]
  d/dt(a^(-1)(t)) = [1, 0, (1/5)t^(-4/5), 1/2]

Hi! To calculate a^(-1)(t) and d/dt(a^(-1)(t)), follow these steps:

1. Write down the given function a(t): a(t) = [t, 0, t^5, 2t]

2. Calculate the inverse function a^(-1)(t) by swapping the roles of x and y (in this case, t and the function itself): a^(-1)(t) = [t, 0, t^(1/5), t/2]

3. Calculate the derivative of a^(-1)(t) with respect to t:
  d/dt(a^(-1)(t)) = [d/dt(t), d/dt(0), d/dt(t^(1/5)), d/dt(t/2)]

4. Compute the derivatives:
  d/dt(t) = 1
  d/dt(0) = 0
  d/dt(t^(1/5)) = (1/5)t^(-4/5)
  d/dt(t/2) = 1/2

5. Write the final answer:
  a^(-1)(t) = [t, 0, t^(1/5), t/2]
  d/dt(a^(-1)(t)) = [1, 0, (1/5)t^(-4/5), 1/2]

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A basis for W⊥ is {(2, 6, 1, 0), (0, -2, 0, 1)}. To find a basis for the orthogonal complement W⊥ of the subspace W spanned by w1 and w2, we need to find all vectors that are orthogonal to both w1 and w2.

Let v = (x, y, z, w) be a vector in W⊥. Then we have the following two equations:

w1 · v = 0

w2 · v = 0

where "·" denotes the dot product. Substituting the given vectors and the components of v, we get the following system of linear equations:

x - y + 4z - 2w = 0

y - 3z + w = 0

We can solve this system of equations to find an equation for the plane that contains all vectors orthogonal to W. Adding the two equations, we get:

x - 2z = 0

Solving for x, we get x = 2z. Then substituting into the first equation, we get:

y = 6z - 2w

So a vector v in W⊥ can be written as v = (2z, 6z - 2w, z, w) = z(2, 6, 1, 0) + w(0, -2, 0, 1).

Therefore, a basis for W⊥ is {(2, 6, 1, 0), (0, -2, 0, 1)}.

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The fractional coverage from Company A  is 40 % and the fractional coverage for Company B is 60 %.

Company A pays $340,000 and Company B pays $510,000 of the total loss.

Fractional coverage from Company A = Coverage from Company A / Total Coverage

= $ 500, 000 / ( 500, 000 + 750, 000 )

= $ 500, 000 / $ 1, 250, 000

= 0. 4

= 40%

Fractional coverage from Company B = Coverage from Company B / Total Coverage

= $ 750, 000 / $ 1, 250,000

= 0. 6

= 60 %

Amount paid by Company A = Fractional coverage from Company A x Total Loss

= 0.4 x $ 850,000

= $ 340, 000

Amount paid by Company B = Fractional coverage from Company B x Total Loss

= 0. 6 x  $ 850,000

= $ 510,000

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The coefficient of x5y8 in (x+y)13 is 1287. This is the answer obtained by using the binomial theorem and the formula for binomial coefficients.

The binomial theorem states that (x+y)n = ∑j=0n (nj) xn−j yj, where (nj) = n! / j! (n-j)! is the binomial coefficient.

To find the coefficient of x5y8 in (x+y)13, we need to find the term where j = 8, since xn−j yj = x5y8 when n = 13 and j = 8.

The coefficient of this term is then (n j) = (13 8) = 13! / 8! 5! = 1287. This means that x5y8 is multiplied by 1287 in the expansion of (x+y)13.

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The zeros of the function G(t) are given by t = -1 + √(20.25g) and t = -1 - √(20.25g).

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

To find the zeros of the function G(t), we need to find the values of t that make G(t) equal to zero. So, we start by setting G(t) to zero and solving for t:

G(t) = 0

(t+1)2 - 20.25g = 0 [substituting G(t) in place of 0]

(t+1)2 = 20.25g [adding 20.25g to both sides]

t+1 = ±√(20.25g) [taking the square root of both sides]

t = -1 ± √(20.25g) [subtracting 1 from both sides]

So, the zeros of the function G(t) are given by t = -1 + √(20.25g) and t = -1 - √(20.25g).

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The formula for calculating the 95% confidence interval for the difference in proportions is: (p1 - p2) ± 1.96 * sqrt{ [p1(1 - p1) / n1] + [p2(1 - p2) / n2] } where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and 1.96 is the z-score for the 95% confidence level.

In this scenario, we are interested in comparing the proportions of people who made donations when contacted by telephone and when they received first-class mail requests. We have two independent samples, each of size 200, and we know the proportion of people who made donations in each sample.

We can use the formula mentioned above to calculate the 95% confidence interval for the difference in proportions. The formula takes into account the sample sizes, sample proportions, and the z-score for the desired confidence level.

The confidence interval provides a range of values for the true difference in proportions between the two groups. If the confidence interval includes zero, we cannot reject the null hypothesis that the difference in proportions is zero, meaning there is no significant difference between the two groups. If the confidence interval does not include zero, we can conclude that there is a significant difference in the proportions between the two groups.

In summary, the formula mentioned above can be used to calculate the 95% confidence interval for the difference in proportions between two independent samples, which provides insight into whether there is a significant difference between the two groups.

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No, failing to reject the null hypothesis does not mean that we have proved it to be true beyond all doubt.

The null hypothesis is simply a statement that we assume to be true until we have sufficient evidence to reject it. Failing to reject the null hypothesis means that we do not have enough evidence to reject it, but it does not necessarily mean that the null hypothesis is true. There could be other factors or sources of variation that we have not accounted for in our analysis, which could affect our conclusion.

a. The null hypothesis is that the mean IQ for college students is 90, and the alternative hypothesis is that the mean IQ is less than 90.

b. The test statistic is:

[tex]t = ( \bar x -\mu) / (\sigma / \sqrt{n} )[/tex]

where [tex]\bar x[/tex] is the sample mean, [tex]\mu[/tex] is the hypothesized population mean, σ is the population standard deviation, and n is the sample size. Plugging in the given values, we get:

t = (84 - 90) / (18 / √61) = -2.48

c. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. Since the alternative hypothesis is one-tailed (less than), we look for the area to the left of the observed t-value in the t-distribution with 60 degrees of freedom. Using a t-table or a calculator, we find the p-value to be 0.0082.

d. At the 0.05 level of significance, the p-value (0.0082) is less than the level of significance, so we reject the null hypothesis. This means that we have sufficient evidence to conclude that the mean IQ for college students is less than 90.

e. Based on the sample of 61 college students, we have sufficient evidence to conclude that the mean IQ for college students is less than 90. This suggests that the professor's initial claim of a mean IQ of 90 for college students may not be accurate.

The complete question is:

If we fail to reject the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer.

A professor claims that the mean IQ for college students is 90. He collects a random sample of 61 college students to test this claim and the mean IQ from the sample is 84.

a. What are the null and alternative hypotheses to test the initial claim?

b. Compute the test statistic. Assume the population standard deviation of IQ scores for college students is 18 points.

c. Find the p-value to test the claim at the 0.05 level of significance. Show/explain how you found these values.

d. Find a conclusion for the test (i.e., reject or fail to reject the null hypothesis). State your reasoning (i.e., why?).

e. Interpret your conclusion from part (d) by putting your results in context of the initial claim.

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The absolute value of the determinant of the matrix formed by the given sides is 3, which represents the volume of the paralleled pipe.

To find the volume of a parallelepiped with three sides given as vectors, we take the triple scalar product (also known as the box product) of the vectors.

Let's first find the three vectors given in the problem statement:

Now we take the triple scalar product:

a · (b x c) = a · d

where d = b x c is the cross product of b and c.

b x c = det([[j,k], [3, -1]])i - det([[i,k], [6,-1]])j + det([[i,3], [6,2]])k

= (-3i - 7j - 18k)

So, d = b x c = -3i - 7j - 18k

Now,

a · d = (1)(-3) + (0)(-7) + (0)(-18) = -3

Thus, the volume of the parallelepiped is |-3| = 3 cubic units.

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From the given figure the angle x° is quals to 38°.

Given triangle is a right-angled triangle,

In the right-angled triangle the opposite side of the triangle = 6.1

The hypotenuse of the triangle = 9.9

In a right-angled triangle, by using little big trigonometry we know that,

sin theta = opposite side of the triangle/hypotenuse side of the triangle

From the given figure sin x° = opposite side of x / hypotenuse side

sin x° = 6.1/9.9

x° = [tex]sin^{-1}[/tex]  (6.1/9.9)

x° = 38.03°

From the above analysis, we can conclude that the angle of x° is equal to 38.03° ≅ 38°.

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