What is the prerimiter of ABC with a angle of 29 side length of 10 and angle of 61

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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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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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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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 approximate demand when the price is decreased to $2.25 is 4,800 boxes.

We know that demand is inversely proportional to price, so we can set up the following equation:

x ∝ 1/p

If x denotes demand and p denotes price. We can introduce a proportionality constant k to this equation:

x = k/p

Using the information in the problem, we can calculate the value of k. The demand is 1,800 boxes when the price is $6:

1,800 = k/6

Solving for k, we get:

k = 6 x 1800

k = 10,800

Now we can use this value of k to find the demand when the price is $2.25:

x = 10,800/2.25

x ≈ 4,800

Therefore, the approximate demand when the price is $2.25 is 4,800 boxes.

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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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Therefore, it will take approximately 11.67 years for the money to grow from $10,000 to $15,000 at an annual interest rate of 4.5% compounded continuously.

To solve this problem, we can use the continuous compounding formula: A = Pe^(rt), where A is the final amount, P is the initial principal, e is Euler's number (approximately 2.71828), r is the annual interest rate as a decimal, and t is the time in years. We want to find t when A = $15,000 and P = $10,000. Plugging in these values and solving for t, we get:
$15,000 = $10,000e^(0.045t)
1.5 = e^(0.045t)
ln(1.5) = 0.045t
t = ln(1.5)/0.045
t ≈ 11.67 years

Therefore, it will take approximately 11.67 years for the money to grow from $10,000 to $15,000 at an annual interest rate of 4.5% compounded continuously.

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a. The x-intercepts of the function f(x) = x² + 4x - 12 are (-6, 0) and (2, 0).

b. The y-intercept of the function f(x) = x² + 4x - 12 is (0, -12)

c. The minimum of the function f(x) = x² + 4x - 12 is -16.

In Mathematics and Geometry, the x-intercept of the graph of any function simply refers to the point at which the graph of a function crosses or touches the x-coordinate and the y-value of "f(x)" is equal to zero (0).

Part a.

By critically observing the graph representing the function f(x), we can logically deduce the following x-intercept:

When y = 0, the x-intercept of f(x) are (-6, 0) and (2, 0).

Part b.

By critically observing the graph representing the function f(x), we can logically deduce the following y-intercept:

When x = 0, the y-intercept of f(x) is equal to (0, -12).

Part c.

By critically observing the graph representing the function f(x), we can logically deduce that it has a minimum value of -16.

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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 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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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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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 minimum sample size is  n = 97

The minimum sample size is define as the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power.

We have the information from the question is:

The margin of error is  E = 1.25

The  standard deviation is  s = 7.5

The confidence level is  90% then the level of significance is mathematically represented as:

[tex]\alpha =100-90[/tex]

[tex]\alpha =10%[/tex]%

[tex]\alpha =0.10[/tex]

Now, The critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table  

The value is [tex]Z_\frac{\alpha }{2}=1.645[/tex]

The minimum sample size is mathematically evaluated as:

[tex]n=\frac{Z_\frac{\alpha }{2}(s^2)}{E^2}[/tex]

Plug all the values in above formula:

[tex]n=\frac{1.645^2(7.5)^2}{1.25^2}[/tex]

After calculation, we get :

n = 97

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The given question is incomplete, complete question is:

What is the minimum sample size required to estimate a population mean with 90% confidence when the desired margin of error is 1.25? The standard deviation in a pre-selected sample is  7.5

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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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 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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The lenght is the circumference of that 3in circle.

the circumference of a circle = 2[tex]\pi[/tex]r

= 2*3*[tex]\pi[/tex]

= 6[tex]\pi[/tex]

So the area of the rectangle = 8.2 * 6[tex]\pi[/tex]

= 49.2[tex]\pi[/tex]

Pick the last answer

The area of the label is 49.2π square inches.

To find the area of the label, we need to calculate the area of the rectangle. The formula to calculate the area of a rectangle is A = length × width. In this case, the length of the rectangle is 8.2 inches, which matches the height of the cylinder. The width of the rectangle is equal to the circumference of one of the circles, which can be calculated using the formula C = 2πr, where r is the radius of the circle. Since the radius is 3 inches, the circumference is 2π(3) = 6π inches. Therefore, the area of the rectangle, which is the label, is 8.2 × 6π = 49.2π square inches.

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The total surface area that Lisa would need to cover is 2 times the sum of the areas of the top and bottom faces plus 2 times the sum of the areas of the side faces.

To calculate the total surface area that Lisa would need to cover, we need to consider all the faces of the block.

Assuming the block is a rectangular prism, it will have six faces: a top face, a bottom face, and four side faces.

Let's denote the length, width, and height of the block as L, W, and H, respectively.

Top and Bottom Faces:

The top and bottom faces have dimensions of L × W each, so their combined area is 2 × (L × W).

Side Faces:

The side faces consist of four rectangles, two with dimensions of L × H and two with dimensions of W × H. So, the combined area of the side faces is 2 × (L × H) + 2 × (W × H).

Now, we can calculate the total surface area by summing up the areas of all the faces:

Total Surface Area = 2 × (L × W) + 2 × (L × H) + 2 × (W × H)

Therefore, the total surface area that Lisa would need to cover is 2 times the sum of the areas of the top and bottom faces plus 2 times the sum of the areas of the side faces.

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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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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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To convert polar coordinates (r,θ) to rectangular coordinates (x,y). Therefore, the rectangular coordinates of the point with polar coordinates (1,116) are approximately (-0.211, 0.978).

we use the following formulas:

x = r cos(θ)

y = r sin(θ)

In this case, the polar coordinates are (1,116). Therefore, we have:

r = 1

θ = 116°

Converting θ from degrees to radians, we get:

θ = 116° * π/180 = 2.025 radians

Substituting these values into the formulas above, we get:

x = r cos(θ) = 1 cos(2.025) ≈ -0.211

y = r sin(θ) = 1 sin(2.025) ≈ 0.978

Therefore, the rectangular coordinates of the point with polar coordinates (1,116) are approximately (-0.211, 0.978).

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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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Step-by-step explanation:

A semi-circle is half of a circle. Therefore, to find the area of a semi-circle, you just have to find the area of a full circle and then divide it by two. It will be faster than you think.[1]

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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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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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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Aaron added 6 fish to his tank.

To find out how many fish Aaron added to his tank, we can subtract the initial number of fish from the final number of fish.

Final number of fish - Initial number of fish = Number of fish added

13 fish - 7 fish = 6 fish

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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  smallest such $n$ is $12$.

To solve the problem, we can start by expanding $(n+r)^3$ and approximating it by ignoring the term $r^3$, since $r$ is small.

We  then want to find the smallest positive integer $n$ such that there exists a positive real number $r$ less than $1/1000$ satisfying the equation. We can try different values of $n$ starting from $n=1$ and incrementing by $1$ until we find a value of $n$ that works.

By  testing a few values, we find that $n=12$ works, giving us $1728 + 1296r + 324r^2$, which is less than $(12+1/40)^3$. Therefore, the smallest such $n$ is $12$.

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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"