Jacki has just completed this table using a rule for changing x into y. Which rule was she using?

The value of the integral ∫[4,-2]f(x)8 dx is -576.

If the average value of a continuous function f on the interval [−2, 4] is 12, it means that the definite integral of f(x) over the interval [−2, 4] is equal to 12 times the length of the interval, which is 6. In other words:

∫[-2,4]f(x) dx = 12(4-(-2)) = 72

Now we can use this information to evaluate the integral ∫[4,-2]f(x)8 dx. We can use the constant multiple rule of integration to pull out the constant 8 from the integral:

∫[4,-2]f(x)8 dx = 8 ∫[4,-2]f(x) dx

Since we already know that ∫[-2,4]f(x) dx = 72, we can evaluate the integral over the interval [4,-2] using the property of definite integrals that says:

∫[a,b]f(x) dx = -∫[b,a]f(x) dx

Therefore:

∫[4,-2]f(x) dx = -∫[-2,4]f(x) dx = -72

Substituting this value back into the original expression, we get:

∫[4,-2]f(x)8 dx = 8 ∫[4,-2]f(x) dx = 8(-72) = -576

Therefore, the value of the integral ∫[4,-2]f(x)8 dx is -576.

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The area of the rectangle based on the information given is A = 2 / (x - 3)

Length: (x² + 7x + 12) / (5x² - 45)

We can factor the numerator and denominator to simplify:

(x + 3)(x + 4) / 5(x - 3)(x + 3)

The (x + 3) factors cancel out, leaving:

(x + 4) / 5(x - 3)

Width: (10x - 10) / (x² + 3x - 4)

We can factor the denominator to simplify:

(x + 4)(x - 1)

So the width becomes:

10(x - 1) / (x + 4)(x - 1)

The (x - 1) factors cancel out, leaving:

10 / (x + 4)

Now we can multiply the length and width to get the area:

[(x + 4) / 5(x - 3)] * [10 / (x + 4)]

The (x + 4) factors cancel out, leaving:

2 / (x - 3)

The area of the rectangle is:A = 2 / (x - 3)

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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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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 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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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]

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 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 work done by force f in moving a particle counterclockwise around the given curve is zero.

The given force [tex]f = (x-5y)i + (5x-y)j[/tex] is a conservative force. A conservative force is characterized by having a curl of zero, indicating that it can be derived from a potential function. For conservative forces, the work done over a closed path is always zero.

In this case, the particle moves counterclockwise around the circle defined by [tex](x-4)^2 + (y-4)^2 = 16[/tex]. Since the force f is conservative, the work done by f on the particle is independent of the path taken and depends only on the initial and final positions of the particle.

As the particle completes one counterclockwise revolution around the circle, it returns to its initial position. Therefore, the work done by force f is zero, indicating that the force does not transfer any energy to or from the particle.

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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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The slant height of the cone is 24 cm. 

We are given that the surface area of the cone is 252.58 cm' with the radius of 6 cm.

The surface area of the cone is calculated through the equation as;

                   A = πr² + πrl

where r is the radius and l is the slant height.

Now Substituting the known values to the equation,

             252.58 = π(6 cm)² + π(6 cm)(l)

l = 24 cm.

Thus, the slant height of the cone is 24 cm. 

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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 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 solution is: The General Solution of the expression is: Cos x = 2/3

Here, we have,

Explanation:

Given expression:

3sinx = 2tanx​

Find:

Simplify expression......

Computation:

By taking given trigonometric equation

⇒ 3sinx = 2tanx​

Using substitution method...

⇒ sinx = [2/3]tax

⇒ sinx / tanx = 2/3

⇒ sinx / [sinx / cosx] = 2/3

⇒ [sinx × cosx] / sinx = 2/3

⇒ cosx = 2/3

Simplify expression

Cos x = 2/3

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complete question:

Determine the general solution of 3sin x=2tan x​

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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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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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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One possible square root of 340 modulo 437 is 281.

To find a square root of a number modulo another number, we can use the Tonelli-Shanks algorithm. First, we check that 340 is a quadratic residue modulo 437 by calculating 340^((437-1)/2) mod 437, which gives 1. Then we can choose a starting value for the square root, such as 281, and use the Tonelli-Shanks algorithm to iteratively improve our estimate. After a few iterations, we should arrive at a more precise square root. Note that there may be multiple square roots, and this is just one example.

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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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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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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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Using an equation, the commission percentage (x) that you earn if you work 8 hours with sales of $1,400 and total earnings of $176 for the day is 4%.

The commission percentage can be determined by equating the normal earnings for the day and the sales commission to the total earnings.

An equation represents two or more mathematical expressions using the equal symbol.

The hourly rate of earnings = $15

The number of hours worked for the day = 8 hours

The commission percentage = x

The Sales for the day = $1,400

The total earnings for the day = $176

Total earnings = hourly rate x 8 hours + 1,400x

176 = 15 x 8 + 1,400x

176 = 120 + 1,400x

56 = 1,400x

x = 0.04

x = 4%

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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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Answer:

6

Step-by-step explanation:

8+9+1 is 18, and since there are 3 numbers, we must divide 18 by 3, 2 which is 6

Answer: 6

Step-by-step explanation:

8+9+1 = 18
3 numbers
18/3 = 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 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 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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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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To represent the relationship between the height of the waterfall and the height of its picture on a postcard, we can use the equation y = 7n.The correct answer is option A.

Here, 'y' represents the height of the waterfall, and 'n' represents the factor by which the waterfall is taller than the picture.

The equation states that the height of the waterfall, 'y', is equal to 7 multiplied by 'n'. Since the picture on the postcard is 7 cm tall, multiplying it by 'n' gives us the height of the waterfall.

For example, if the factor 'n' is 3, the height of the waterfall would be 7 cm * 3 = 21 cm. If 'n' is 5, the height would be 7 cm * 5 = 35 cm. The equation allows us to calculate the height of the waterfall for any given factor 'n'.

Using this equation, we can determine the height of the waterfall based on the height of its picture on the postcard, providing a simple and direct relationship between the two.

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The probable question may be:

A waterfall is n times the height of its picture on a postcard. Which equation represents the height, y, of the waterfall if the picture is 7 cm tall?

A. y = 7n

B. y = 7 + n

C. y = n - 7

D. y = 7 / n