Jhon bought a rectangular door mat that was 1/2 meter long and 3/10 meter wide. What is the area of the door mat ?

Answer: 55%

Step-by-step explanation:

To find the percentage of 11/20, we can use the following formula:

Percentage = (Numerator ÷ Denominator) × 100

Substituting the values from 11/20 into the formula, we get:

Percentage = (11 ÷ 20) × 100

Percentage = 0.55 × 100

Percentage = 55%

Therefore, the percentage of 11/20 is 55%.

Answer:

Solution: 11/20 as a percent is 55%

Step-by-step explanation:

First, convert the fraction into a decimal by dividing the numerator by the denominator:

11/20 = 0.55

If we multiply the decimal by 100, we will get the percentage:

00.5 * 100 = 55

We can see that 11/20 is percentage is 55.

Percentage of lifetime cost that was interest = $407.

Let's begin by finding the monthly payment that Hillary made to pay off her laptop over two and a half years.

If she paid off the $804 balance with identical monthly payments, then the total amount she paid is equal to the balance plus the interest:

Total amount paid = balance + interest

We can use the formula for the present value of an annuity to solve for the monthly payment, where PV is the present value (in this case, $804), r is the monthly interest rate (which we can find from the APR), n is the total number of payments (30 months), and PMT is the monthly payment:

PV = PMT * (1 - (1 + r)^(-n)) / r

We can solve this equation for PMT:

PMT = PV * r / (1 - (1 + r)^(-n))

The monthly interest rate is the annual percentage rate divided by 12, and the number of payments is the number of years times 12:

r = 0.1127 / 12 = 0.009391667

n = 2.5 * 12 = 30

Using these values, we get:

PMT = 804 * 0.009391667 / (1 - (1 + 0.009391667)^(-30)) = $33.00

So Hillary made 30 monthly payments of $33.00 to pay off her laptop.

Next, we can calculate the cost of electricity over six years. There are 365 days in a year, and 2 leap years in the six-year period, for a total of 6*365+2 = 2192 days.

At $0.27 per day, the total cost of electricity is:

2192 * $0.27 = $592.64

Now we can calculate the total cost of the laptop over six years.

Hillary paid $33.00 per month for 30 months, or a total of 30 * $33.00 = $990.00. She also paid $592.64 for electricity. Therefore, the total cost of the laptop is:

$990.00 + $592.64 = $1582.64

The interest she paid on her credit card is the difference between the total amount she paid and the cost of the laptop:

Interest = Total amount paid - Cost of laptop

Interest = $990.00 + interest on $804 balance - $804 - $592.64

Simplifying this expression, we get:

Interest = $185.36 + interest on $804 balance

To find the interest on the $804 balance, we can use the formula for compound interest, where P is the principal (in this case, $804), r is the annual interest rate (11.27%), and t is the time in years (2.5 years):

A = P*(1 + r/n)^(n*t)

Here, we can set the number of compounding periods per year, n, to 12 since the interest is compounded monthly. Substituting the given values, we get:

A = $804*(1 + 0.1127/12)^(12*2.5) = $1026.12

So the interest on the $804 balance is:

Interest on $804 balance = $1026.12 - $804 = $222.12

Plugging this value into our expression for Interest, we get:

Interest = $185.36 + $222.12 = $407.48

Finally, we can find the percentage of the lifetime cost of the laptop that was interest:

Percentage of lifetime cost that was interest = Interest / Total cost of laptop * 100%

Percentage of lifetime cost that was interest = $407.

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Answer:

.15(180) - .10(180) = .05(180) = 9

9 more people voted for split pea soup than for potato soup.

Answer: True

Step-by-step explanation:

Answer:

Its 12

Step-by-step explanation:

1.   Following PEMDAS, we first solve the equation inside the parentheses.

       -4(-2) = 8

       -12 ÷ 3 = -4

       -20 + 4 = -16

2.    Now, we have the following expression:

       8 + (-4) + (-16)

 3.  Again, following PEMDAS, we solve the equation inside the parentheses first.

       8 + (-4) + (-16) = 8 - 4 - 16

  4. Finally, we solve the equation from left to right.

       8 - 4 - 16 = 8 - (4 + 16)

       8 - (20) = -12

Therefore, the value of the expression is -12.

If two prisms are similar, their corresponding dimensions are proportional.

Let's assume that the ratio of the corresponding lengths of the smaller prism to the larger prism is k:1, where k is a constant.

Then the ratio of the corresponding surface areas of the smaller prism to the larger prism is [tex](k^2):1[/tex], because the surface area of a prism is proportional to the square of its length.

In this problem, the surface area of the smaller prism is not given.

However, we can find the ratio of the corresponding lengths of the smaller prism to the larger prism using the fact that they are similar.

The height of the smaller prism can be found as follows:

[tex]7/5 = h/L[/tex]

where h is the height of the smaller prism and L is the length of the larger prism.

Solving for h, we get:

[tex]h = (7/5)L[/tex]

The ratio of the corresponding lengths of the smaller prism to the larger prism is 7:5.

The ratio of the surface areas of the smaller prism to the larger prism is:

[tex](7/5)^2 : 1 = 49/25 : 1[/tex]

We know that the surface area of the larger prism is [tex]90 m^2.[/tex]

Let's denote the surface area of the smaller prism by A. Then we can set up an equation:

(49/25)A = 90

Solving for A, we get:

A = (25/49) * 90 = 45/7 ≈ 6.4

The surface area of the smaller prism is approximately [tex]6.4 m^2.[/tex]

(Note: The units of the surface area are not provided for the smaller prism, so I assumed the same units as the larger prism.

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This is an example of an experiment. In an experiment, researchers manipulate the independent variable (in this case, the presence or absence of an electrical impulse) to determine its effect on the dependent variable (the number of repetitions the subjects can lift a weight).

The researcher randomly assigned subjects to either receive the electrical impulse or not, which is a key feature of experimental design.

By doing so, the researcher can ensure that any differences observed between the two groups are due to the manipulation of the independent variable, rather than any pre-existing differences between the groups.

In contrast, an observational study merely observes existing characteristics or behaviors of a population, without any manipulation or control of variables.

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(a) The greatest amount of snowfall in any of the cities = 50 inches.

(b) The data is most concentrated in the second quarter (Q₂), ranging from 6 inches to 20 inches.

(c) The data is most spread out in the fourth quarter (Q₄), ranging from 32 inches to 50 inches.

A box plot is a type of graphical representation that summarizes the distribution of a dataset based on the five-number summary: the minimum value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum value.

A box plot consists of a rectangular box, which spans from Q1 to Q3, with a vertical line inside it representing the median. The length of the box represents the interquartile range (IQR), which is the range between Q1 and Q3.

Whiskers, which are lines extending from the top and bottom of the box, indicate the range of the dataset outside of the IQR.  

Here we have

Jyllina created this box plot representing the number of inches of snow that fell this winter in different nearby cities

The data ranges from 4 to 50 inches

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

(a) The greatest amount of snowfall in any of the cities will equal the highest value which is 50 inches.

(b) The data is most concentrated in the second quarter (Q₂), which ranges from 6 inches to 20 inches.

This can be inferred from the fact that the box plot shows the smallest range between the minimum and maximum values, as well as the smallest size of the box, which represents the interquartile range (IQR).

(c) The data is most spread out in the fourth quarter (Q₄), which ranges from 32 inches to 50 inches.

This can be inferred from the fact that the box plot shows the largest range between the minimum and maximum values, as well as the largest size of the box, which represents the IQR.

Additionally, the whiskers, which represent the range of values outside the IQR, are also the longest in this quarter, indicating that there are more extreme values in this range compared to the other quarters.

Therefore,

(a) The greatest amount of snowfall in any of the cities = 50 inches.

(b) The data is most concentrated in the second quarter (Q₂), ranging from 6 inches to 20 inches.

(c) The data is most spread out in the fourth quarter (Q₄), ranging from 32 inches to 50 inches.

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Complete Question:

Frank's monthly credit card payment for consolidating his credit cards will be $467.29.

Option C is the correct answer.

We have,

To calculate the monthly credit card payment for consolidating Frank's credit cards, we can use the formula for the monthly payment on a loan:

[tex]M = P (r (1 + r)^n) / ((1 + r)^n - 1),[/tex]

where M is the monthly payment, P is the total loan amount (sum of all credit card balances), r is the monthly interest rate, and n is the number of months.

First, let's calculate the total loan amount:

Total loan amount = $2,380 + $4,500 + $1,580 + $900 = $9,360.

Next, let's calculate the monthly interest rate:

Monthly interest rate = APR / 12 = 18% / 12 = 1.5%.

Now, let's calculate the monthly payment using the formula:

[tex]M = $9,360 \times (0.015 (1 + 0.015)^{24}) / ((1 + 0.015)^{24} - 1).[/tex]

Using a calculator, we can compute the value of M:

M ≈ $467.286.

Rounding to the nearest cent,

Frank's monthly credit card payment for consolidating his credit cards will be $467.29.

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The part of the surface area equation are

From the question, we have the following parameters that can be used in our computation:

SA = B + 1/2Pl

In the equation, we have

Part 1 = B

Part 2 = P

Part 3 = l

When the above parts are labelled, we have

Part 1 = B = Base area

Part 2 = P = Perimeter of base

Part 3 = l = Height of pyramid

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On babysitting for four hours, the differently earned money will be Quinton would make $5 more than Reema.

Reema's charge for babysitting for four hours = 10 × 4

Reema's charge = $40

For Quinton, keep the value of hours in expression -

Quinton's charge = 5 (4) + 25

Quinton's charge = 20 + 25

Quinton's charge = $45

As Quinton's charge is greater than Reema, she makes more money.

Amount of more money made by Quinton = 45 - 40

Subtract to find the difference

Amount of more money made by Quinton = $5

Hence, correct answer is Quinton would make $5 more than Reema.

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Two possible values of x are 10 and 40.

To find two possible values of x, we need to first determine the highest and lowest numbers in the set.

Highest number: 46
Lowest number: 4

To get a range of 55, we need the highest number minus the lowest number to equal 55.

46 - 4 = 42

But we know that there are 8 numbers in the set, so the range must be spread out over those 8 numbers. To get an idea of how much each number should increase, we can divide 42 by 7 (the number of gaps between the numbers) to get an average increase of 6.

Now we can use this average increase to find two possible values of x.

1) If x is 6 more than the lowest number (4 + 6 = 10), then the set becomes:

4, 9, 10, 32, 35, 38, 46, 46

And the range is:

46 - 4 = 42

2) If x is 6 less than the highest number (46 - 6 = 40), then the set becomes:

4, 9, 35, 38, 40, 46, 46, 32

And the range is:

46 - 4 = 42

Therefore, two possible values of x are 10 and 40.

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Using the intersection of sets we find that A ∩ B = {1, 3}

A set is a collection of well ordered items.

Givent hat set A = {x| x is an od number greater than 0 and less than 10} and set B = {1, 2, 3, 4}, we desire to find A ∩ B. We proceed as follows.

First since set A = {x| x is an odd number greater than 0 and less than 10}, its elements are A = {1, 3, 5, 7, 9}

Also, set B =  {1, 2, 3, 4}

Since we require A ∩ B which is the intersection of both sets and is  the elements that both sets have in common. Since both sets have element 1 and 3 in common, we have that

A ∩ B = {1, 3}

So, A ∩ B = {1, 3}

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Answer:

I can definitely help you with that math problem! Given the information about the bicycle wheel, we need to find the function that models the height of a spot on the edge of the wheel. We know that the wheel has a diameter of 26 inches, which means the radius is half of that, or 13 inches. Isabelle rides the bike so that the wheel makes two complete rotations per second, which means the period of the function is 1/2 second (since it takes half a second for the wheel to complete one rotation).

Using the formula for a sinusoidal function, we can write the function as h(t) = A sin(2π/B (t - h)) + k, where A is the amplitude, B is the period, h is the horizontal shift, and k is the vertical shift. We can determine the values of these parameters as follows:

- Amplitude: The amplitude is half the distance between the highest and lowest points of the function. Since the radius of the wheel is 13 inches, the highest and lowest points are 26 inches apart. Therefore, the amplitude is 13 inches.

- Period: We know that the period is 1/2 second, so B = 2π/1/2 = 4π.

- Horizontal shift: The function starts at its highest point, so there is no horizontal shift. Therefore, h = 0.

- Vertical shift: The center of the wheel is at a height of 13 inches above the ground, so the vertical shift is also 13 inches.

Putting it all together, we get the function h(t) = 13 sin(4πt) + 13, which corresponds to option C. This function models the height of a spot on the edge of the wheel as Isabelle rides the bike. I hope this explanation helps! Let me know if you have any other questions or if there's anything else I can assist you with.

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The second wedge has a greater volume by 4.125 cubic inches. Therefore, the correct option is 4.

To determine which cheese wedge has a greater volume, you need to calculate the volume of each triangular prism using the given dimensions.

For the first wedge:

1. Calculate the area of the base (isosceles triangle):

(base x height) / 2 = (2.5 in x 4.5 in) / 2 = 5.625 square inches

2. Calculate the volume of the prism:

base area x thickness = 5.625 sq in x 3 in = 16.875 cubic inches

For the second wedge:

1. Calculate the area of the base (isosceles triangle):

(base x height) / 2 = (3 in x 4 in) / 2 = 6 square inches

2. Calculate the volume of the prism:

base area x thickness = 6 sq in x 3.5 in = 21 cubic inches

To find which wedge has a greater volume and by how much, subtract the smaller volume from the larger volume:

21 cu in - 16.875 cu in = 4.125 cu in.

Therefore, the correct answer is option 4: The second wedge has a greater volume by 4.125 cubic inches.

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Answer:

Please look at the picture provided to see the correct triangle because each figure has no a,b, or c. Thanks

The distance between points E and B is (2/3)*AB, or (2/3)*(7+x) units.

Since the dotted line is the perpendicular bisector of side AB, it means that it cuts the line AB into two equal halves. Thus, the distance between points E and the dotted line is equal to the distance between point A and the dotted line.

We know that the distance between points E and A is 7 units, and since the dotted line bisects AB, the distance between point A and the dotted line is equal to the distance between point B and the dotted line. Let's call this distance 'x'.

Therefore, we have two equal distances (7 units and 'x') that add up to the length of AB. This means that:

AB = 7 units + x

However, we also know that the dotted line is the perpendicular bisector of AB, meaning that it forms right angles with both A and B. This creates two right-angled triangles, AED and BED, where DE is the perpendicular line from point E to AB.

Using Pythagoras' theorem, we can find the length of DE in terms of 'x':

(DE)² + (AE)² = (AD)²

(DE)² + (7)² = (AB/2)²

(DE)² + 49 = (AB²)/4

(DE)² = (AB²)/4 - 49

(DE)² = (AB² - 196)/4

(DE)² = (x²)/4

DE = x/2

Therefore, the distance between points E and B is equal to the length of DE plus the distance between point B and the dotted line, which is also equal to 'x'. Therefore, the distance between points E and B is:

EB = (x/2) + x = 1.5x

We can substitute this into the equation we found earlier:

AB = 7 units + x

AB = 7 units + (2/3)*EB

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To find the moment of inertia about the​ x-axis of the given area, we can use the formula:

Ix = ∫y^2 dA
Where Ix is the moment of inertia about the​ x-axis and dA is an infinitesimal area element.
First, we need to find the limits of integration. The curve y^2 = 4x - 2 intersects the x-axis at (1/2, 0). Also, the area is bounded by the x-axis and the line x = 7. Therefore, the limits of integration for x are from 1/2 to 7.
Now, we can express the infinitesimal area element as dA = y dx. Also, we can solve the given equation for x in terms of y as x = (y^2 + 2)/4. Therefore, we can write:
Ix = ∫y^2 (y dx)
Ix = ∫[(y^3)/4 + (y/2)] dx, with limits from 1/2 to 7
Ix = [(y^3)/16 + (y^2)/4] evaluated at x = 7 and x = 1/2
Ix = [(49y^3)/16 + (49y^2)/4] - [(y^3)/16 + (y^2)/4]
Ix = (48y^3)/16 + (48y^2)/4
Ix = 3y^3 + 12y^2

To find the moment of inertia about the x-axis, we need to substitute y with x and take the integral from 1/2 to 0 (since the area is in the first quadrant):
Ix = ∫3x^3 + 12x^2 dx, with limits from 1/2 to 0
Ix = [x^4/4 + 4x^3] evaluated at x = 1/2 and x = 0
Ix = (1/64) + 0 - (0 + 0)
Ix = 1/64

Therefore, the moment of inertia about the​ x-axis of the first-quadrant area bounded by the curve y^2=4x−2, the​ x-axis, and x=7 is 0.0156 (rounded to 1 decimal place).
To find the moment of inertia (I_x) about the x-axis of the first-quadrant area bounded by the curve y^2 = 4x - 2, the x-axis, and x = 7, we need to use the following formula:
I_x = ∫(y^2 * dA)

Here, dA represents the differential area element. Since the curve is defined in terms of y^2, let's express y in terms of x:
y = ±√(4x - 2)
As we are considering the first quadrant, we will take the positive root:
y = √(4x - 2)
Now, let's find the differential area element, dA:
dA = y*dx
Substitute the expression for y into dA:
dA = √(4x - 2)*dx

Now, substitute dA into the formula for I_x and integrate with respect to x:
I_x = ∫(y^2 * dA) = ∫((4x - 2) * √(4x - 2)*dx)
Integrate this expression with limits of integration from x = 0 (where the curve intersects the x-axis) to x = 7:
I_x ≈ 203.33

Therefore, the moment of inertia about the x-axis for the given region is approximately 203.3 (rounded to 1 decimal place).

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The triangle has two sides with equal lengths (AB and AC) and one side with a different length (BC). This makes it an isosceles triangle.

Triangle ABC has vertices A(0,0), B(0,3), and C(3,0).

To determine the type of triangle, we can find the lengths of the sides using the distance formula:

AB = sqrt((0-0)^2 + (3-0)^2) = sqrt(0 + 9) = 3

BC = sqrt((3-0)^2 + (0-3)^2) = sqrt(9 + 9) = sqrt(18) = 3√2

AC = sqrt((3-0)^2 + (0-0)^2) = sqrt(9 + 0) = 3

The triangle has two sides with equal lengths (AB and AC) and one side with a different length (BC). This makes it an isosceles triangle.

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a. To find the inverse function of f(x), we need to interchange the roles of x and y and solve for y. So, we have:

y = 1x - 3
x = 1y - 3
x + 3 = y
Therefore, the inverse function of f(x) is f^-1(x) = x + 3.
The domain of f^-1(x) is the range of f(x). Since f(x) = 1x - 3 is a linear function, its domain is all real numbers. Therefore, the range of f(x) is also all real numbers. In interval notation, we can write this as (-inf, inf).

The range of f^-1(x) is the domain of f(x). As we determined in part b, the domain of f(x) is all real numbers. Therefore, the range of f^-1(x) is also all real numbers. In interval notation, we can write this as (-inf, inf).
Hi! I'd be happy to help you with your question.
a. To find the inverse function of f(x) = 1x - 3, you can follow these steps:
1. Replace f(x) with y: y = 1x - 3
2. Swap x and y: x = 1y - 3
3. Solve for y: y = x + 3
So, the inverse function f^(-1)(x) = x + 3.
The domain of f^(-1) refers to the set of all possible x-values. Since the inverse function is a linear function with no restrictions, the domain of f^(-1) is all real numbers. In interval notation, this is written as (-∞, ∞).

c. The range of f^(-1) refers to the set of all possible y-values (output). Again, since it's a linear function with no restrictions, the range of f^(-1) is also all real numbers. In interval notation, this is written as (-∞, ∞).

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The price of the stereo system after the discount and markup is $691.20.

The markup price represents the price after adding a percentage of the discounted price.

The markup can be determined using the markup factor, which increases 100% by the markup percentage.

The discount offered on the stereo system = 20%

Original sales price of the system = $720

Discount factor = 0.8 (1 - 0.2)

Discounted price = $576 ($720 x 0.8)

Markup percentage after the discount = 20%

Markup factor = 1.2 (1 + 0.2)

Marked up price = $691.20 ($576 x 1.2)

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The magnitude of the resultant force is approximately 60.28 newtons. The angle between the resultant force and the smaller force is approximately 50.5 degrees.

To find the magnitude of the resultant force, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the two forces are acting at right angles, so we can treat them as the sides of a right triangle:

resultant force^2 = (39n)^2 + (46n)^2

resultant force^2 = 1521n^2 + 2116n^2

resultant force^2 = 3637n^2

resultant force = sqrt(3637n^2) = 60.28n

So the magnitude of the resultant force is approximately 60.28 newtons.

To find the angle that the resultant force makes with the smaller force, we can use trigonometry.

We know that the two forces are at right angles, so the angle between the resultant force and the smaller force is the same as the angle between the resultant force and the larger force. Let's call this angle θ. Then we have:

tan θ = (larger force) / (smaller force)

tan θ = 46n / 39n

θ = tan^-1(46/39) = 50.5°

Therefore, the angle between the resultant force and the smaller force is approximately 50.5 degrees.

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The number written as a percentage is:

772.5666118%

To do this, just multiply the number by 100%.

For example, for any number A, the percentage form of A is:

p = A*100%

Here the number is  7.725666118, then the percentage form of this number will be:

N =  7.725666118*100% = 772.5666118%

That is the number as a percentage.

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Answer: X= 4/5x + 8

Step-by-step explanation: Distribution factor

The least-squares regression line for predicting the 2016 budget from the 2006 budget is y = 18.18 + 0.9365x, their budgets to differ in 2016 is 9.365 million and the budget for organization is 111.83 million.

In a regression graph, the regression line closest to the data points is shown. By replacing alternative values of x in the regression equation, this statistical technique can assist analyse the behaviour of a dependent variable y when the independent variable x changes.

The corporation can identify the optimal asset price in relation to the cost of capital by using regression. It is used in the stock market to calculate the impact of stock price movements on the price of underlying commodities.

Regression analysis may be used in marketing to understand how pricing fluctuations affect the rise or reduction in products sales. It is particularly successful in forecasting future sales by correlating market factors, weather forecasts, economic situations, and prior sales.

a) Here we find the least square regression line for predicting the 2016 budget :

Regression line y = a+bx

b = [tex]\frac{n(Exy)-(Ex*Ey)}{n(Ex)^r-(Ey)^r}[/tex]

= [tex]\frac{2216256 - 1219050}{2220421-1155625}[/tex]

= [tex]\frac{997206}{1064796}[/tex]

= 0.9365

a= y-bx

= [tex]\frac{Ey}{n} -b\frac{Ex}{n}[/tex]

162 - (0.9635 x 153.5714286)

= 162 - 143.819

= 18.18

y = 18.18 + 0.9365x

b) For 10 million change in 2006, there will be

= 10 x 0.9365 = 9.365 million change in 2016

c) If x = 100,

y = 18.18 + 0.9365(100) = 111.83 million.

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Complete question:

Organization 2006 2016

Corporation for Public Broadcasting 460 445

Institute of Museum and Library Services 247 237

National Endowment for the Humanities 142 148

National Endowment for the Arts 124 148

National Gallery of Art 95 147

Commission of Fine Arts 2 3

Advisory Council on Historic Preservation 5 6

Source: National Endowment for the Arts

a. Compute the least-squares regression line for predicting the 2016 budget from the 2006 budget.

b. If two institutions have budgets that differ by 10 million dollars in 2006, by how much would you predict their budgets to differ in 2016?

c.Predict the 2016 budget for an organization whose 2006 budget was 100 million dollars.

Answer:

Step-by-step explanation:

false i think

Answer:

True

Step-by-step explanation:

Answer:

a + 5b - 7c

Step-by-step explanation:

Let the number to be added by X

then,

a - 2b + 3c + X = 2a + 3b - 4c

X =  a + 5b - 7c

The coordinates of p so that p partitions the segment of AB in the ratio 7 to 2 if A( -5,4) and B( -8,-2) is P(x, y) = [-22/3, -2/3].

In this scenario, line ratio would be used to determine the coordinates of the point P on the directed line segment that partitions the segment into a ratio of 1 to 4.

In Mathematics and Geometry, line ratio can be used to determine the coordinates of P and this is modeled by this mathematical equation:

P(x, y) = [(mx₂ + nx₁)/(m + n)],  [(my₂ + ny₁)/(m + n)]

By substituting the given parameters into the formula for line ratio, we have;

P(x, y) = [(mx₂ + nx₁)/(m + n)],  [(my₂ + ny₁)/(m + n)]

P(x, y) = [(7(-8) + 2(-5))/(7 + 2)],  [(7(-2) + 2(4))/(7 + 2)]

P(x, y) = [(-56 - 10)/(9)],  [(-14 + 8)/9]

P(x, y) = [-66/9],  [(-6)/(9)]

P(x, y) = [-22/3, -2/3]

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Assuming that the trail is a total of "x" miles, we can set up the following equation to solve for "x":

816 = 0.48x

To solve for "x", we can divide both sides by 0.48:

x = 1700

Therefore, the total length of the trail is 1700 miles.

To find the measure of angle 1 when AB is tangent to circle P at point B, we must consider some properties of tangents and circles.

A tangent line to a circle is a line that touches the circle at exactly one point, known as the point of tangency. In this case, line AB is tangent to circle P at point B. A crucial property of tangents is that they are perpendicular to the radius of the circle at the point of tangency. Therefore, the radius PB of circle P is perpendicular to tangent AB at point B.

Now, let's examine angle 1. If angle 1 is the angle formed by the tangent AB and the radius PB at point B, then it is a right angle due to the aforementioned property. In this case, the measure of angle 1 is 90 degrees.

However, if angle 1 is not directly formed by the tangent and radius, more information is needed to determine its measure. For example, if angle 1 is an angle inside the circle, you would need to know the measure of other angles or lengths of chords within the circle to calculate it.

In summary, if angle 1 is formed by tangent AB and radius PB at point B, its measure is 90 degrees because tangents are perpendicular to the radius at the point of tangency. If angle 1 is not formed by the tangent and radius, additional information is required to determine its measure.

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