The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 2.
x = 3, y = 12
x = 3/7, y = 7/6

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 volume V of the solid S can be found using the formula for the volume of a cone, which is V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.

In this case, the height of the cone is 3h and the base radius is 3r, so we have r = 3r and h = 3h. Substituting these values into the formula, we get:

V = (1/3)π(3r)^2(3h)

V = 27πr^2h

Therefore, the volume of the solid S is 27πr^2h.

To explain further, we can see that the solid S is a cone with a height that is three times the given height and a base radius that is three times the given radius. The volume of any cone can be found using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. By substituting the given values of r and h into this formula, we can obtain the volume of the cone. In this case, we have to substitute 3r for r and 3h for h to get the volume of the solid S in terms of the given height and radius. Simplifying the expression, we get the final answer as 27πr^2h.

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Note that the probability of randomly selecting one person from this table and the person being a female is 50 %.

Given that we have 10 high school girls and 10 women.

Therefore, the total number of females is 10 ( girls)   + 10 (women) =  20.

For the  total number of people in the sample, we have 10 high school boys, 10 high school girls, 10 men, and 10 women, which  sums up to 10 + 10 + 10 + 10 = 40.

Probability = Number of females / Total number of people

= 20 / 40

= 0.5 or 50%

Thus, the probability of randomly selecting one person from the table and the person being a female is 50%.

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81 can be written as 3^4 and 64 can be written as 2^6. There are several ways to write 81 and 64 using exponents. For example:

- 81 = 3^4 = (3^2)^2

- 81 = 9^2/3 = (3^2)^2/3

- 81 = (27/3)^2 = 27^(2/3)

- 64 = 2^6 = (2^3)^2

- 64 = 4^3/2 = (2^2)^3/2

- 64 = (8/2)^3 = 8^(3/2)

To write 64/81 using exponents, we can use the fact that a fraction can be written as a negative exponent. Therefore,

- 64/81 = 64 * 81^(-1) = 2^6 * 3^(-4) = 2^6/3^4

- 64/81 = (2/3)^(-6/4) = (2/3)^(-3/2)

- 64/81 = 8^2/9^2 = (8/9)^2

These are some examples of how to write 64/81 using exponents. It is important to note that there are infinitely many ways to write a number using exponents, but some forms may be more useful or appropriate in certain situations.

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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 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 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

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]

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 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 solution of the two system of equations using elimination method is x = 8, and y = 3.

The solution of the two system of equations using elimination method is calculated as follows;

The given equations;

2x - 5y = 1   -------- (1)

-3x + 2y = -18 ----- (2)

To eliminate x, multiply equation (1)  by 3 and equation (2) by 2, and add the to equations together;

3:   6x  -  15y = 3

2:   -6x  + 4y  = -36

-----------------------------------

             -11y = -33

               y = 33/11  =  3

Now, solve for the value of x by substituting the value of y back into any of the equations.

2x - 5y = 1

2x - 5(3) = 1

2x - 15 = 1

2x = 16

x = 8

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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 of the triangle is given by the formula:

A =  (3/2)x² + 11x - 8

Which is the one in option b.

Remember that for a triangle of base B and height H, the area is given by the formula:

A = B*H/2

Here we know that the height is:

H = 8 + x

The base is:

B = 3x - 2

Then the area of the triangle is:

A = (8 + x)*(3x - 2)/2

A = (24x - 16 + 3x² - 2x)/2

A = (3x² + 22x - 16)/2

A = (3/2)x² + 11x - 8

Then the correct option is B (though the constant term is -8 instead of -9)

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The enlarged photograph will be 30 inches in width and a proportional length.

Marine has a rectangular photograph that measures 6 inches in width. If she wants to enlarge the photograph using a scale of 1:5, this means that the size of the photograph will be increased by a factor of 5 in both width and height.

To determine the new width of the photograph, we can multiply the original width of 6 inches by 5, which gives us a width of 30 inches for the enlarged photograph.

It's important to note that while the width of the photograph will increase, the aspect ratio of the photograph will remain the same.

This means that the length of the photograph will also be increased by the same factor of 5.

Therefore, the enlarged photograph will be 30 inches in width and a proportional length.

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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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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 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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Note that the first method has the smaller Mean Absolute Error (MAE) which is 1.75

The mean absolute error (MAE) is defined as the average variance between the significant values in the dataset and the projected values in the same dataset.

The steps to finding the MAE in eah case is

For the 1st method

Absolute error (AE) for day 1 = |23 - 23| = 0

AE for day 2 = |5 - 10| = 5

AE for day 3 = |14 - 15| = 1

AE  for day 4 = |20 - 19| = 1

so MAE = (0 + 5 + 1 + 1) / 4 = 1.75

For the 2nd system,

Absolute error (AE) 1 = |20 - 23| = 3

AE for day 2 = |13 - 10| = 3

AE  for day 3 = |14 - 15| = 1

AE for day 4 = |20 - 19| = 1

So the  MAE = (3 + 3 + 1 + 1) / 4 = 2

We can conclude, hence, that t the first method has the smaller   MAE of 1.75.

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The given power series is ∑n=1infinity/n! and we need to find its convergence set. This series converges for -∞ < x < ∞.

The given power series is ∑n=1infinity/n!. To find its convergence set, we can use the ratio test. Applying the ratio test, we get:

lim [n→∞] |(2^(n+1) x^(n+1))/(n+1)!| / |(2^n x^n)/n!|

= lim [n→∞] (2x)/(n+1)

= 0

Since the limit is less than 1 for all values of x, the series converges for all values of x. Therefore, the convergence set is (-∞, ∞).

Intuitively, we can see that since the terms of the series involve a factorial in the denominator, the terms become smaller and smaller as n increases, making it easier for the series to converge. In addition, the term 2^n in the numerator increases rapidly, which can balance out the effect of the denominator. As a result, the series converges for all values of x.

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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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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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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"

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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Using Euler's method with a time step of 0.3, the first two approximations for the solution to the initial value problem y'(t) = t + y, y(0) = 5 are u1 = 6.5 and u2 = 8.035.

Using Euler's method, we can approximate the solution to the initial value problem y'(t) = t + y, y(0) = 5 with a time step of Δt = 0.3 as follows:

At t = 0, y = 5

Using the formula: y1 = y0 + f(y0,t0)Δt, where f(y,t) = t + y

y1 = 5 + (0 + 5)0.3 = 6.5

Using the formula again with y1 and t1 = Δt:

y2 = 6.5 + (0.3 + 6.5)0.3 = 8.035

Thus, the first two approximations u1 and u2 are 6.5 and 8.035, respectively.

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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 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 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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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  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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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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