Determine the equation of the directrix of r = 26. 4/4 + 4. 4 cos(theta) A. X = -6 B. Y = 6 C. X = 6

The total cost of the two tickets is $70.88.

A discount is a reduction in the original price of an item or service. It is often used as a marketing strategy to increase sales by making the product more affordable or attractive to consumers. The amount of the discount is typically expressed as a percentage of the original price.

In the given question,

The cost of one ticket after a 25% discount is:

45 * 0.75 = $33.75

The cost of two tickets is:

2 * 33.75 = $67.50

Adding the 5% sales tax:

67.50 * 1.05 = $70.88

Therefore, the total cost of the two tickets is $70.88.

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Step-by-step explanation:

the gradient = the slope = the incline = ...

many different names for the same thing.

however you call it, it is the ratio

y coordinate change / x coordinate change

whet going from one point on the line to another.

for questions like this we should look for points with integer coordinates (going through a vertex of the coordinate grid squares).

I see for example right at the left beginning (0, 1).

the next one is then (4, 2).

when going from (0, 1) to (4, 2) :

x changes by +4 (from 0 to 4).

y changes by +1 (from 1 to 2).

so, the slope or gradient is

+1/+4 = 1/4

The approximate length of the base of the triangle is 5 units

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. The area of a triangle is expressed as;

A = 1/2 bh

The hexagon is divided into six equal triangles.

The area of the hexagon is 65 units²

Therefore, the area of the triangle = 65/6

= 10.83

Area of a triangle = 1/2 b h

height = 4.3 units

10.83 = 1/2 × 4.3 × b

4.3b = 10.83 × 2

4.3b = 21.66

divide both side by 4.3

b = 21.66/4.3

b = 5.04

therefore the approximated value of the base is 5 units.

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Fred purchased more 2. 45 liters of laundry detergent than bleach is false statement, Fred took 450 milliliters more fabric softener than bleach is false, Fred placed 220 milliliters more laundry detergent than bleach is true, Fred has taken 0. 45 liters more fabric softener than bleach is false.

Fred in total bought 5 liters of liquid laundry detergent which is equal to 5000 milliliters. Then he  bought 3,250 milliliters of fabric softener and 2.8 liters of bleach which is equal to 2800 milliliters.
Fred purchased 45 milliliters more fabric softener than bleach. This statement is false because Fred bought 250 milliliters less fabric softener than bleach.

Fred bought 2.45 liters more laundry detergent than bleach. This statement is false because Fred bought 2.2 liters more laundry detergent than bleach.

Fred has taken 450 milliliters more fabric softener than bleach. This statement is false because Fred bought 250 milliliters less fabric softener than bleach.
Fred on the event of taking 220 milliliters more laundry detergent than bleach is considered a true statement because Fred bought 2200 milliliters more laundry detergent than bleach.

Fred on the event of taking 0.45 liters more fabric softener than bleach is considered a false statement due to the fact that Fred bought 0.25 liters less fabric softener than bleach.

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Sarah wants to attend a private college with a yearly tuition of $31,000. Room and board costs are estimated to be $12,000 per year, and the cost of books and supplies is estimated to be $2,000. To calculate the total cost of her four-year degree, follow these steps:

1. Add the yearly costs together: $31,000 (tuition) + $12,000 (room and board) + $2,000 (books and supplies) = $45,000 per year.
2. Multiply the yearly cost by the number of years in the degree program: $45,000 * 4 = $180,000.

Assuming she receives no financial aid, it will cost Sarah $180,000 to get a four-year degree from this private college.

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The surface area of the cone is approximately 793.10 cm².

What is cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

What is area of cone?

The surface area of a cone is given by the formula:

Surface Area = πr² + πrℓ

where r is the radius of the circular base and ℓ is the slant height of the cone.

To find the surface area of a cone, we need to find the area of its base and the area of its lateral surface and then add them together.

The formula for the lateral surface area of a cone is given by:

L = πrℓ

where r is the radius of the base and ℓ is the slant height of the cone.

The formula for the area of the base of a cone is given by:

B = πr²

where r is the radius of the base.

Given the slant height ℓ = 19 and the radius r = 9, we can find the lateral surface area of the cone as follows:

L = πrℓ

= π(9)(19)

≈ 538.63 cm²

Next, we can find the area of the base of the cone as follows:

B = πr²

= π(9)²

≈ 254.47 cm²

Therefore, the surface area of the cone is:

A = B + L

≈ 793.10 cm²

So, the surface area of the cone is approximately 793.10 cm².

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The divergence of the vector field div((2[tex]x^2[/tex]L - sin(xz)) i + 5j - (sin(xz)) k) at all points where it is defined is: div(F) = 4x - z*cos(xz) - x*cos(xz).

To find the divergence of the given vector field, we need to apply the divergence operator to the vector field.

The divergence operator is given by the following formula:
div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
Where F = (Fx, Fy, Fz) is the vector field.
Let's apply this formula to the given vector field:
F = (2[tex]x^2[/tex] - sin(xz)) i + 5j - (sin(xz)) k
div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
      = (4x - zcos(xz)) + 0 + (-xcos(xz))
Therefore, the divergence of the given vector field is:
div(F) = 4x - zcos(xz) - xcos(xz)
This expression gives the divergence of the vector field at all points where it is defined.
In this case, the vector field is defined for all values of x, y, and z, so the divergence is defined for all points in space.
It is worth noting that the divergence of a vector field represents the rate at which the vector field flows out of a small volume of space surrounding a point.

If the divergence is positive, the vector field is flowing out of the volume; if it is negative, the vector field is flowing into the volume and if it is zero, the vector field is not flowing into or out of the volume.

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The divergence of the given vector field is div(F) = 4x - z × cos(xz) - x × cos(xz).

To find the divergence of the given vector field div( (2[tex]x^2[/tex] - sin(xz)) i + 5j - (sin(xz)) k), follow these steps:

Identify the components of the vector field:

F(x, y, z) = (2[tex]x^2[/tex]- sin(xz), 5, -sin(xz))

Compute the partial derivatives with respect to each variable:

∂F1/∂x = ∂(2[tex]x^2[/tex] - sin(xz))/∂x

∂F2/∂y = ∂(5)/∂y

∂F3/∂z = ∂(-sin(xz))/∂z

Calculate each partial derivative:

∂F1/∂x = 4x - z × cos(xz)

∂F2/∂y = 0

∂F3/∂z = -x × cos(xz)

Add the partial derivatives to find the divergence:

div(F) = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z

div(F) = (4x - z × cos(xz)) + 0 + (-x × cos(xz))

Simplify the expression:

div(F) = 4x - z × cos(xz) - x × cos(xz)

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Answer: D)-3/4

Step-by-step explanation: Rate of change of y with respect to x implies the following formula...(y2-y1)/(x2-x1)...which is also the gradient of the line, you can substitute the value in accordance with the points already ahown in the grid to get, (7-1)/(-4-4)=-3/4

it is 16 steps from second base to third base .we can get this answer by

solving the logic given in the question .we need to add up the number of steps

what is  add up  ?

"Add up" means to calculate the total of two or more numbers or quantities by combining them together. It involves the mathematical operation of addition, which is the process of finding the sum of two or more numbers. For example, if you add up the numbers 3, 5, and 7, t

In the given question,

To find out how many steps it is from second base to third base, we need to add up the number of steps Gavin took in each direction.

Starting at second base, he took 18 steps forward to get caught, and then took 7 steps back. This leaves him 11 steps forward from second base.

Next, he took 3 more steps forward, for a total of 14 steps forward from second base. But then he took 5 steps back, leaving him 9 steps forward from second base.

Then he took 11 more steps forward, for a total of 20 steps forward from second base. But then he took 4 steps back, leaving him 16 steps forward from second base.

Finally, he was tagged out halfway between second and third base. Since he was 16 steps forward from second base, and halfway between second and third base, we can assume that third base is 16 steps away from second base.

Therefore, it is 16 steps from second base to third base.

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Answer:

Step-by-step explanation:

You want the median of the Blue box plot and the IQR of the Pink box plot shown in the figure.

A box plot has 5 vertical lines. From left to right, they are ...

Based on the above, we can see the median is the middle line inside the box of the box plot.

The median of the blue plot is 5.2.

The "Interquartile range" is the difference between the 3rd quartile and the 1st quartile. That is, it is the difference between the two ends of the box, the length of the box.

The IQR of the pink plot is 6.4 -5.6 = 0.8.

__

Additional comment

This is basically an exercise in understanding the terms associated with a box plot, and reading a graph. The only calculation involved is figuring out the missing graph coordinates and finding the difference of the coordinates at the ends of the pink plot.

The local maximum is at x = -9 with a value of 2575, and the local minimum is at x = 1 with a value of -17.

To find the local minimum and maximum of the given function, we need to find its critical points, where the derivative is zero or undefined.

[tex]f(x) = 2x^3 + 30x^2 - 54x + 5[/tex]

[tex]f'(x) = 6x^2 + 60x - 54[/tex]

Setting f'(x) = 0, we get:

[tex]6x^2 + 60x - 54 = 0[/tex]

[tex]x^2 + 10x - 9 = 0[/tex]

(x + 9)(x - 1) = 0

x = -9 or x = 1

Now, we need to determine if these critical points correspond to local minimum or maximum.

To do so, we can use the second derivative test. We calculate the second derivative of f(x):

f''(x) = 12x + 60

At x = -9:

f''(-9) = 12(-9) + 60 = -48 < 0

This means that f(x) has a local maximum at x = -9.

At x = 1:

f''(1) = 12(1) + 60 = 72 > 0

This means that f(x) has a local minimum at x = 1.

To find the values of the local minimum and maximum, we plug in the corresponding x-values into the original function:

[tex]f(-9) = 2(-9)^3 + 30(-9)^2 - 54(-9) + 5 = 2575[/tex]

[tex]f(1) = 2(1)^3 + 30(1)^2 - 54(1) + 5 = -17[/tex]

Therefore, the local maximum is at x = -9 with a value of 2575, and the local minimum is at x = 1 with a value of -17.

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The fraction, David used 13/8 cups of blueberries for the pancakes.

Let's solve it step-by-step using the given information.

1. David has 3 1/2 cups of blueberries.
2. He uses 1/4 cup for a breakfast smoothie.
3. He uses 1/2 of the remaining blueberries for pancakes.

Step 1: Calculate the remaining blueberries after making the smoothie.
3 1/2 - 1/4 = (7/2) - (1/4)

To subtract the fractions, they need a common denominator, which in this case is 4.

(7/2) * (2/2) - (1/4) = (14/4) - (1/4) = 13/4 cups

Step 2: Calculate the amount of blueberries used for the pancakes.
David uses 1/2 of the remaining blueberries for the pancakes, so:

(13/4) * (1/2) = 13/8 cups

David uses 13/8 cups of blueberries for the pancakes.

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The probability that the first survey indicates four children in the family and the second survey indicates one child in the family is 1/50.

We have,

To find the probability of the first survey indicating four children in the family and the second survey indicating one child in the family, we need to consider the number of surveys that fit this condition and divide it by the total number of possible surveys.

According to the table, the number of surveys indicating four children in the family is 8, and the total number of surveys is:

= 9 + 18 + 22 + 8 + 3 = 60.

Since the first survey is returned to the stack and can be chosen again, the probability of the first survey indicating four children in the family is 8/60.

For the second survey, there are 9 surveys indicating one child in the family (as the first survey is returned to the stack and can be chosen again), and the total number of surveys remains 60.

Therefore, the probability of the second survey indicating one child in the family is 9/60.

To find the probability of both events occurring, we multiply the individual probabilities:

Probability = (8/60) x (9/60) = 72/3600 = 1/50

Thus,

The probability that the first survey indicates four children in the family and the second survey indicates one child in the family is 1/50.

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The complete question:

Number of children in family     Number of surveys

one                                                      9

tqo                                                     18

three                                                  22

four                                                      8

five or more                                        3

When x = 5 and dx = 0.1, the differential dy is approximately 96.375.

When x = 5 and dx = 0.2, the differential dy is approximately 192.75.

We can use the chain rule to find the differential of y with respect to x. Let u = 5x + 3, then y = tan(u).

Using the chain rule, we have:

dy/dx = dy/du * du/dx

Taking the derivative of y with respect to u, we have:

dy/du = sec^2(u)

Substituting u = 5x + 3, we have:

dy/du = sec^2(5x + 3)

Taking the derivative of u with respect to x, we have:

du/dx = 5

Substituting x = 5 and dx = 0.1, we have:

dy = dy/du * du/dx * dx

= sec^2(5(5) + 3) * 5 * 0.1

= 192.75 * 0.5

= 96.375

Therefore, when x = 5 and dx = 0.1, the differential dy is approximately 96.375.

Substituting x = 5 and dx = 0.2, we have:

dy = dy/du * du/dx * dx

= sec^2(5(5) + 3) * 5 * 0.2

= 192.75 * 1

= 192.75

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Answer: 8

Step-by-step explanation:

okay so do 17 x 17 - 15 x 15

then to that result do :

squared root.

A restaurant meal cost $90 and there was a 6% meals tax on the price of the meal. The meal tax was $5.40 and the total price of the meal with the meal tax was $95.40.

Find the meal tax, you need to calculate 6% of the $90 meal cost.
Convert the percentage to a decimal by dividing by 100, so 6% = 0.06.
Multiply the meal cost by the decimal. In this case, $90 × 0.06 = $5.40.
So, the meal tax was $5.40.
Find the total price of the meal with the meal tax, you need to add the meal tax to the original cost of the meal.
Add the meal tax to the original cost, so $90 + $5.40 = $95.40.
The total price of the meal with the meal tax was $95.40.

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ABC4 base 16 is equal to 43972 in base ten (decimal).

In order to represent any given number, numerals might be numbers, symbols, figures, or sets of figures.

To convert the number ABC4 base 16 to a numeral in base ten (decimal), we can use the positional notation system. Each digit in the number represents a power of 16, starting from the rightmost digit.

The rightmost digit is 4, which represents 4 x 16⁰ = 4 x 1 = 4.

The next digit is C, which represents 12 (since C is equivalent to the decimal number 12), and it is in the second position from the right. So the value of the second digit is 12 x 16¹ = 12 x 16 = 192.

The next digit is B, which represents 11, and it is in the third position from the right. So the value of the third digit is 11 x 16² = 11 x 256 = 2816.

The leftmost digit is A, which represents 10, and it is in the fourth position from the right. So the value of the fourth digit is 10 x 16³ = 10 x 4096 = 40960.

Now we can add up the values of each digit to get the decimal equivalent of the number:

4 + 192 + 2816 + 40960 = 43972

Therefore, ABC4 base 16 is equal to 43972 in base ten (decimal).

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The missing terms in the blanks  and number of terms  are determined below.

The missing terms in the blanks for the pattern and number of terms  can be determined as follows:

1. 32, 30, 28, 26, 32__________________________, 0

The difference is 2. Thus, subtract 2 till you reach 0. That is:

32, 30, 28, 26, 24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0

Number of terms: 16

2. 75, 70, 65, 60, 55, 50, ____________________, 0

The difference is 5. Thus, subtract 5 till you reach 0. That is:

75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 0

Number of terms: 15

3. 30, 27, 24, 21, ___________________________, 0

The difference is 3. Thus, subtract 3 till you reach 0. That is:

30, 27, 24, 21, 18, 15, 12, 9, 6, 3, 0

Number of terms: 10

4. 81, 72, 63, ______________________________, 0

The difference is 9. Thus, subtract 9 till you reach 0. That is:

81, 72, 63, 54, 45, 36, 27, 18, 9, 0

Number of terms: 9

5. 48, 44, 40, 36, __________________________, 0

The difference is 4. Thus, subtract 4 till you reach 0. That is:

48, 44, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4, 0

Number of terms: 12

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The fare for this flight would be: $225.

To determine the fare for the flight that was 350 miles with a residual of -105, we need to use the least-squares regression equation:

ý = 102.50 + 0.65x

where ý is the predicted fare and x is the distance in miles.

We know that the distance for this flight is 350 miles, so we can substitute x = 350 into the equation:

ý = 102.50 + 0.65(350)

ý = 102.50 + 227.50

ý = 330

Therefore, the predicted fare for this flight is $330.

The residual of -105 means that the actual fare for this flight was $105 less than the predicted fare based on the regression line. Therefore, the actual fare for this flight would be:

$330 - $105 = $225

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The measure of arc CAD is either 240 or 260.

Without additional information, it is not possible to determine the measure of arc CAD with certainty. The measure of an arc depends on the central angle that subtends it.

If the central angle is known, the measure of the arc can be calculated using the formula: measure of arc = (central angle / 360) x circumference of the circle. However, without knowing the central angle, we cannot determine the measure of arc CAD.

Therefore, we need to be provided with additional information such as the measure of another angle that is related to the central angle, or the length of a chord that subtends the arc in order to determine the central angle and the measure of arc CAD.

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The limit is equal to 10. We didn't need to use L'Hospital's rule or any other advanced method, as the limit was easily evaluate through simplification and direct substitution.

We can simplify the expression as follows:

[tex]lim x→5 (x + 5) x = lim x→5 (10) = 10[/tex]

Now, we can directly evaluate the limit by substituting 5 for x:

[tex]lim x→5 (x + 5) x = lim x→5 (10) = 10[/tex]

Therefore, the limit is equal to 10. We didn't need to use L'Hospital's rule or any other advanced method, as the limit was easily evaluatable through simplification and direct substitution.

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The Stafford loan will have a lower balance by $272.54 at the time of repayment.

For the Stafford Loan, the principal is $10,720 and the interest rate is 5.95% compounded monthly, so the monthly interest rate is 0.0595/12 = 0.00495833.

After 12 months, the balance of the loan will be:

$10,720(1 + 0.00495833)¹² = $11,204.60

After the time of six-month grace period, the balance will be:

$11,204.60(1 + 0.00495833)⁶ = $11,689.66

For the PLUS Loan, the principal is $11,443.63 and the interest rate is 6.55% compounded monthly, so the monthly interest rate is 0.0655/12 = 0.00545833.

After 12 months, the balance of the loan will be:

$11,443.63(1 + 0.00545833)¹² = $11,962.20

Therefore, the Stafford Loan will have a lower balance at the time of repayment by:

$11,962.20 - $11,689.66 = $272.54

Hence, the Stafford loan will have a lower balance by $272.54 at the time of repayment.

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The value of x is 2.8 The scale factor of left polygon to the right polygon, GKNM to RPTV is 1.4.

Since the two polygons GKNM and RPTV are similar, their corresponding sides are proportional. Thus, we can set up the following proportion

(GK + KN + NM)/GK = (RP + PT + TV + VR)/RP

Plugging in the given values and simplifying, we get

(8.4 + 3x - 2 + 4)/8.4 = (x + 5 + 3 + 3 + 6.3)/(x + 5)

15.4/(3x + 2.4) = (x + 17.3)/(x + 5)

Cross-multiplying and solving for x, we get

x = 2.8

To find the scale factor of GKNM to RPTV, we can divide the corresponding side lengths

(GK + KN + NM)/(RP + PT + TV + VR) = (8.4 + 3(2.8) - 2 + 4)/(2.8 + 5 + 3 + 3 + 6.3) = 1.4

Therefore, the scale factor of GKNM to RPTV is 1.4.

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The solution to the differential equation with the given initial condition is:

[tex]u = -1/2 * ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

To solve equation, we can separate the variables and write:

[tex]1/e^{(2u)} du = e^{(9t)} dt[/tex]

Integrating both sides, we get:

[tex]\int 1/e^{(2u)} du = \int e^{(9t)} dt[/tex]

Integrating the left side requires the substitution v = 2u, dv/du = 2, and du = dv/2, which gives:

[tex]\int 1/e^{(2u)} du = \int 1/2 * 1/e^v dv = -1/2 * e^{(-2u)}[/tex]

Integrating the right side gives:

[tex]\int e^{(9t)} dt = 1/9 * e^{(9t)}[/tex]

Substituting these integrals back into the original equation, we get:

[tex]-1/2 * e^{(-2u)} = 1/9 * e^{(9t)} + C[/tex]

where C is the constant of integration.

We can solve for the constant of integration using the initial condition u(0) = 4:

[tex]-1/2 * e^{(-24)} = 1/9 * e^{(90)} + C[/tex]

[tex]-1/2 * e^{(-8)} = 1/9 + C[/tex]

[tex]C = -1/2 * e^{(-8)} - 1/9[/tex]

Therefore, the solution to the differential equation [tex]du/dt = e^{(2u+9t)}[/tex] with initial condition u(0) = 4 is:

[tex]-1/2 * e^{(-2u)} = 1/9 * e^{(9t)} - 1/2 * e^{(-8)} - 1/9[/tex]

Solving for u, we get:

[tex]e^{(-2u)} = -2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9[/tex]

Taking the natural logarithm of both sides, we get:

[tex]-2u = ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

Dividing by -2, we get:

[tex]u = -1/2 * ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

Therefore, the solution to the differential equation with the given initial condition is:

[tex]u = -1/2 * ln(-2/9 * e^{(9t)} + 4/9 * e^{(-8)} + 2/9)[/tex]

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The mechanic used 412 - 411.375 = 0.625 gallons of motor oil during the day.

To solve this problem, we need to convert both measurements to the same unit.

1 gallon = 8 pints (since there are 8 pints in a gallon)

So the mechanic started with:

412 gallons * 8 pints/gallon = 3,296 pints

And ended with:

5 pints

To find how much motor oil the mechanic used during the day, we can subtract the ending amount from the starting amount:

3,296 pints - 5 pints = 3,291 pints

To convert this back to gallons, we divide by 8:

3,291 pints / 8 pints/gallon = 411.375 gallons (rounded to three decimal places)

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The equation is y = x + 3 which shows the relationship of the two activities.

The equation to show the relationship between the number of hours Julie practices her violin each week (y) and the number of hours she studies (x) is:

y = x + 3

This means that the number of hours Julie practices her violin each week is equal to the number of hours she studies, plus three additional hours.

A linear equation is an equation that represents a straight line on a graph. It can be written in the form:

y = mx + b

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There are different ways a person can make up a story about a word problem. The square root of 81 - 7 is 2. Hence the story is given below.

The hypothetical story goes like this: Mia was trying to figure out the length of the missing side of a square garden bed. She knew that the total area of the garden bed was 81 square feet. She also knew that she had already planted in a section of the garden bed that was 7 feet wide.

So, what will be the length of the missing side of the garden bed if Mia wants to make sure it is completely filled with soil. In order to solve this problem, Mia used the equation: the square root of 81-7=2. She plugged in the values she knew and solved for the missing side length. Therefore, The solution will be:= ✓81 - 7= 9 - 7= 2

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A story for the word problem is that what will be the length of the missing side of a square garden when the total area is 81 feet² and the width was 7 feet wide?

From the information, we are seeking a story problem that has the equation had the square root of 81 - 7 = 2.

This will be written as story for the word problem is that what will be the length of the missing side of a square garden when the total area is 81 feet ² and the width was 7 feet wide?

The length will be calculated thus :

= √ 81 - 7

= 9 - 7

= 2

The length of the missing side is 2 feet.

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The fair price for annual premiums for the policy holders to pay is $288.42

To determine the fair price for annual premiums, we need to take into account the company's expenses and desired profit. The company's total expenses can be calculated as the sum of the claims paid, marketing expenses, and labor expenses:

Total expenses = Claims paid + Marketing expenses + Labor expenses

Total expenses = $38000 + $1750 + $6000

Total expenses = $45750

To calculate the fair price for annual premiums, we need to add the desired profit to the total expenses and divide by the number of policy holders:

Fair price = (Total expenses + Desired profit) / Number of policy holders

Fair price = ($45750 + 20% of $45750) / 190

Fair price = ($45750 + $9150) / 190

Fair price = $54900 / 190

Fair price = $288.42

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Using the Maclaurin polynomial of degree 6, the approximation for the integral ∫³√(1+x²)dx from 0 to 0.4 is ≈ 0.41721.

To find the Maclaurin polynomial of degree 6 for the function g(x) = ³√(1+x²), we will use the binomial series expansion:

(1+x)^(n) = 1 + nx + (n(n-1)x²)/2! + (n(n-1)(n-2)x³)/3! + ...

In our case, n = 1/3, and x = x²:

g(x) = (1+x²)^(1/3) = 1 + (1/3)x² - (1/9)(2/3)x⁴/2! + (1/27)(2/3)(-1/3)x⁶/3! + ...

Now, we can write the Maclaurin polynomial of degree 6:

g(x) ≈ 1 + (1/3)x² - (1/27)x⁴ + (2/729)x⁶

To approximate the integral, we can integrate the polynomial from 0 to 0.4:

∫(1 + (1/3)x² - (1/27)x⁴ + (2/729)x⁶)dx from 0 to 0.4 ≈ [x + (1/9)x³ - (1/135)x⁵ + (1/2187)x⁷] evaluated from 0 to 0.4

Now, plug in the limits:

≈ [0.4 + (1/9)(0.4³) - (1/135)(0.4⁵) + (1/2187)(0.4⁷)] - [0 + (1/9)(0³) - (1/135)(0⁵) + (1/2187)(0⁷)]

≈ 0.4 + 0.01778 - 0.00059 + 0.00002

≈ 0.41721

Thus, using the Maclaurin polynomial of degree 6, the approximation for the integral ∫³√(1+x²)dx from 0 to 0.4 is ≈ 0.41721.

This can be evaluated using basic integration techniques to get an approximate value of the integral. This method is useful for approximating integrals that cannot be solved exactly, and the accuracy of the approximation can be improved by using higher degree Maclaurin polynomials.

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