The rectangular model is made up of squares each square is a equal size what percent of the model shaded

The evaluated missing length in right triangle by using the Pythagorean theorem is  9 yards under the condition given the triangle is a right triangle.

The Pythagoras theorem projects that in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides,
It is given to us that in a right triangle,
Hypotenuse = 15 yd
Perpendicular = 12 yd

Therefore, applying Pythagoras theorem;
Base² = 15² - 12²

Base² = 225 - 144

Base² = 81

Base = √81

Base = 9 yards

Hence, The missing length present in the right triangle by applying the Pythagorean theorem is,
9 yards

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The complete question is

Determine the missing length in each right triangle using the Pythagorean theorem. Round the answer to the nearest tenth, if necessary

The possible value of <J is 78 degrees

It is important to note that the different trigonometric identities are;

Also, the law of sines in a triangle is expressed as;

sin A/a = sin B/b = sin C/c

Given that the angles are in capitals and the sides are in small letters.

From the information given, we have that;

sinI/i = sin J/j

Substitute the values, we get;

sin 72 /70 = sin J/72

cross multiply the values, we have;

sin J = 68. 476/70

divide the values

sin J = 0. 9782

Find the inverse of sin

<J = 78 degrees

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The slope of the line that divides the region into two equal parts is 8/7.

We begin by finding the x-coordinates of the points where the parabola intersects the x-axis. Setting y = 0, we get:

[tex]2x - 7x^2 = 0[/tex]

x(2 − 7x) = 0

x = 0 or x = 2/7

Thus, the parabola intersects the x-axis at x = 0 and x = 2/7.

We want to find the slope of the line through the origin that divides the region bounded by the parabola and the x-axis into two regions with equal area.

Let's call this slope m.

We know that the area under the parabola from x = 0 to x = 2/7 is:

A = ∫[0,2/7] (2x − 7[tex]x^2[/tex]) dx

A = [[tex]x^2[/tex] − (7/3)[tex]x^3[/tex]] from 0 to 2/7

A = (4/21)

Since we want the line to divide this area into two equal parts, the area to the left of the line must be (2/21).

Let's call the x-intercept of the line h. Then the equation of the line is y = mx, and the area to the left of the line is:

(1/2)h(mx) = (1/2)mhx

We want this to be equal to (2/21), so we can solve for h:

(1/2)mhx = (2/21)

h = (4/21m)

The x-coordinate of the point of intersection of the line and the parabola is given by:

2x − 7[tex]x^2[/tex] = mx

Simplifying, we get:

[tex]7x^2 - (2 + m)x = 0[/tex]

Using the quadratic formula, we get:

[tex]x = [(2 + m) \pm \sqrt((2 + m)^2 - 4(7)(0))]/(2(7))[/tex]

x = [(2 + m) ± √(4 + 4m + [tex]m^2[/tex])]/14

x = [(2 + m) ± (2 + m)]/14

x = 1/7 or x = −(2/7)

Since we want the line to pass through the origin, we must choose x = 1/7, and we can solve for m:

[tex]2(1/7) - 7(1/7)^2 = m(1/7)[/tex]

m = 8/7

Therefore, the slope of the line that divides the region into two equal parts is 8/7.

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The difference between the two banks is you will make $367 more at Wells Fargo than at TD Bank, to determine which bank has the better savings program, let's compare the total interest earned at TD Bank and Wells Fargo.

TD Bank offers a 6% interest rate for 7 years, while Wells Fargo offers a 5% interest rate for 9 years. If you deposit $10,000, we can calculate the total interest earned at each bank using the formula for compound interest:

A = P(1 + r/n)^(nt)

Where A is the future value of the investment, P is the principal amount ($10,000), r is the annual interest rate, n is the number of times interest is compounded per year, t is the number of years, and "^" denotes exponentiation.

Assuming annual compounding (n = 1), the calculations for each bank are as follows:

TD Bank:
A = $10,000(1 + 0.06/1)^(1*7)
A = $10,000(1.06)^7
A = $15,018.93

Wells Fargo:
A = $10,000(1 + 0.05/1)^(1*9)
A = $10,000(1.05)^9
A = $15,386.16

Comparing the two banks, Wells Fargo's savings program will make you the most money, with a future value of $15,386.16. The difference between the two banks is:

Difference = Wells Fargo - TD Bank
Difference = $15,386.16 - $15,018.93
Difference = $367.23

Rounding to the nearest dollar, you will make $367 more at Wells Fargo than at TD Bank. Therefore, Wells Fargo has the better savings program in this scenario.

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The answer is reasonable because it is positive and also the equation is true . it would take Lara about 5.3 hours to clean the tank by herself.

Let's denote the time it takes for Lara to clean the tank alone as "L". We can use the formula for the combined work rate of two people, which is:

(1/7) + (1/L) = (1/3)

Multiplying both sides by the least common denominator, 21L, gives:

3L + 21 = 7L

Subtracting 3L from both sides, we get:

21 = 4L

Dividing both sides by 4, we get:

L = 5.25 hours (to the nearest tenth)

The answer is reasonable because it is positive, and it is also less than 7 hours, which is Kevin's time. When substituted back into the original equation, we get:

(1/7) + (1/5.25) = (1/3)

0.1429 + 0.1905 = 0.3333

0.3334 ≈ 0.3333

The equation is true, so the answer is reasonable. Therefore, it would take Lara about 5.3 hours to clean.

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The instantaneous rate of change of cost with respect to x when x = 100 is 4.

We can begin by calculating C(100+h) and C(100):

C(100+h) = 0.000002(100+h)^3 + 4(100+h) + 300

C(100+h) = 0.000002(1,000,000 + 300h^2 + 30h^2 + h^3) + 400 + 4h + 300

C(100+h) = 0.000002h^3 + 0.0006h^2 + 4h + 700

C(100) = 0.000002(100)^3 + 4(100) + 300

C(100) = 2 + 400 + 300

C(100) = 702

Therefore,

C(100+h) - C(100) = (0.000002h^3 + 0.0006h^2 + 4h + 700) - 702

C(100+h) - C(100) = 0.000002h^3 + 0.0006h^2 + 4h - 2

Now, we can find the rate of change of cost with respect to x by dividing this expression by h and taking the limit as h approaches 0:

(C(100+h) - C(100))/h = (0.000002h^3 + 0.0006h^2 + 4h - 2)/h

(C(100+h) - C(100))/h = 0.000002h^2 + 0.0006h + 4 - (2/h)

As h approaches 0, the term 2/h approaches infinity, which means the rate of change of cost with respect to x is undefined. However, we can calculate the limit of the expression as h approaches 0 from the left and from the right to see if it has a finite value:

limit (h->0+) ((C(100+h) - C(100))/h) = 4

limit (h->0-) ((C(100+h) - C(100))/h) = 4

Since the left and right limits are equal, the overall limit exists and equals 4. Therefore, the instantaneous rate of change of cost with respect to x when x = 100 is 4.

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The probability that a randomly selected 10th grade boy exceeds 70 in is approximately 0.0127 or 1.27%.

The height of a 16-year-old boy in the 96th percentile is approximately 73 inches.

For the first question, we need to find the z-score for a height of 70 inches using the formula:

z = (x - μ) / σ

where x is the height of 70 inches, μ is the mean of 63.5 inches, and σ is the standard deviation of 2.9 inches.

z = (70 - 63.5) / 2.9 = 2.241

Using a standard normal table, we can find the area to the right of this z-score, which represents the probability that a randomly selected 10th grade boy exceeds 70 inches. The area to the right of 2.24 is 0.0127. Therefore, the probability is approximately 0.0127 or 1.27%.

For the second question, we need to find the z-score associated with the 96th percentile using a standard normal table. The 96th percentile is the point below which 96% of the data falls and above which 4% of the data falls. This corresponds to a z-score of approximately 1.75.

To find the height of a 16-year-old boy in the 96th percentile, we can use the formula:

x = μ + z * σ

where x is the height we want to find, μ is the mean of 68.3 inches, σ is the standard deviation of 2.9 inches, and z is the z-score we just found.

x = 68.3 + 1.75 * 2.9 = 73.28

Therefore, the height of a 16-year-old boy in the 96th percentile is approximately 73 inches.

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The velocity of the velociraptor is 0.4 m/s [E] - 0.8 m/s [W] + 0.3 m/s [S] and  the speed of the velociraptor is 2 m/s.

To calculate the speed of the velociraptor, we need to divide the total distance traveled by the total time taken:

Total distance = 5m + 10m + 3m + 2m = 20m

Total time = 10 seconds

Speed = Total distance / Total time

     = 20m / 10s

     = 2m/s

To calculate the velocity of the velociraptor, we need to consider both the magnitude of its speed and its direction. We can calculate the displacement by subtracting the final position from the initial position:

Displacement = (5m [E] + 10m [W] + 3m [S] + 2m [W])

           = (5m [E] - 8m [W] + 3m [S])

           = 5m [E] - 8m [W] + 3m [S]

Note that we have used negative sign for the distance traveled towards the west.

The time taken is 10 seconds.

Velocity = Displacement / Time taken

        = (5m [E] - 8m [W] + 3m [S]) / 10s

        = 0.4 m/s [E] - 0.8 m/s [W] + 0.3 m/s [S]

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Josh downloaded 6 songs, which corresponds to option B.

The given equation is:

c = 1.390 + 3.50d

Where c represents the total cost and d represents the number of songs downloaded. You mentioned that Josh spent a total of $20.18. So, we'll set c to 20.18 and solve for d:

20.18 = 1.390 + 3.50d

Step 1: Subtract 1.390 from both sides of the equation:
20.18 - 1.390 = 3.50d
18.79 = 3.50d

Step 2: Divide both sides of the equation by 3.50:
18.79 / 3.50 = d
5.36857 = d

Since d must be a whole number (as you can't download a fraction of a song), we round it down to the nearest whole number:

d = 5

However, 5 is not among the given options. This indicates there may be a typo in the question. If the correct equation is:

c = 0.390 + 3.50d

Then, solving for d with the given total cost of $20.18:

20.18 = 0.390 + 3.50d

Step 1: Subtract 0.390 from both sides:
20.18 - 0.390 = 3.50d
19.79 = 3.50d

Step 2: Divide both sides by 3.50:
19.79 / 3.50 = d
5.65429 = d

Rounding to the nearest whole number:

d = 6

Thus, Josh downloaded 6 songs, which corresponds to option B.

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The total length of the string in feet is 6 feet, and the combined capacity of both water jugs in cups is 75 cups.

Step 1: To find the total length of the string in inches, Mara needs to add the lengths of the three pieces of string:

30 inches + 22 inches + 20 inches = 72 inches

So the total length of the string in inches is 72 inches.

Step 2: To convert inches to feet, we need to divide the number of inches by 12 (since there are 12 inches in a foot):

72 inches ÷ 12 = 6 feet

Therefore, once Mara puts together the three pieces of string, the total length will be 6 feet.

Step 1: To find out how many ounces of water both water jugs hold altogether, we need to add the capacity of the two jugs:

300 ounces + 300 ounces = 600 ounces

So both water jugs together can hold 600 ounces of water.

Step 2: To convert ounces to cups, we need to divide the number of ounces by 8 (since there are 8 ounces in a cup):

600 ounces ÷ 8 = 75 cups

Therefore, both water jugs together can hold 75 cups of water.

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The cafe served 12 regular coffees.

Out of the 30 coffees served at Molly's cafe this morning, 40% were regular coffee. To determine the number of regular coffees, we can calculate 40% of 30.

To find the value,

Step 1: Convert the percentage to a decimal by dividing it by 100. So, 40% = 40/100 = 0.4.
Step 2: Multiply the total number of coffees served by the decimal. So, 30 * 0.4 = 12.

Hence, the cafe served 12 regular coffees. The remaining 60% (or 18 coffees) would be decaffeinated. It is important to note that percentages represent proportions or fractions of a whole. In this case, 40% indicates that 40 out of 100 parts (or 40/100) are regular coffees. By applying this proportion to the total number of coffees served (30), we can determine the specific quantity. This method can be used in various scenarios involving percentages to find a portion of a whole. Therefore, Molly's cafe served 12 regular coffees and 18 decaffeinated coffees, making a total of 30 coffees.

Your answer: Molly's cafe served 12 regular coffees this morning.

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The situation in which Shawn's activity will result in a final value of zero is Shawn travels east ten feet and then travels south ten feet. The correct option is D.

This is because when Shawn travels east ten feet, he moves horizontally to the right of his starting point. When he travels south ten feet after that, he moves vertically downwards from his previous position, cancelling out the horizontal movement he made earlier.

The displacement caused by Shawn's movement in the east direction is equal in magnitude but opposite in direction to the displacement caused by his movement in the south direction.

The net displacement of Shawn's movement is zero, and he ends up back at his starting point. Options A, B, and C do not involve any movements that result in a net displacement of zero.

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Answer: $532,000

Step-by-step explanation:

If the company is making $140,000 and they get 20% each year, we just multiply it by 20%, or 0.20, and get $28,000. So, we would multiply that by 14, the years the company operated, and then add it to the original $140,000.

28,000 x 14 = 392,000

392,000 + 140,000 = 532,000

So, over the course of 14 years, the company made a profit of $532,000.

The number of cells found in an average Koala is 1.30 x 10¹², under the condition that a cell is a sphere with radius 10 or 0. 001 centimeter.

Then the volume of a sphere with radius 10 cm is considered to be 4/3π(10)³ cubic cm that is approximately 4,188.79 cubic cm.
The evaluated volume of a sphere with radius 0.001 cm is 4/3π(0.001)³ cubic cm that is approximately 0.00000419 cubic cm.

Then the evaluated mass of a single cell is  found by applying the formula
mass = density x volume
In case of larger cell, the mass will be
mass = 1.1 g/cm³ x 4,188.79 cubic cm
= 4,607.67 grams

In case of  smaller cell, the mass will be

mass = 1.1 g/cm³ x 0.00000419 cubic cm
= 0.00000461 grams

As koalas measure an average of 6 kilograms or 6,000 grams², we can evaluate the number of cells in an average koala using division of the weight of the koala by the mass of a single cell

In case of larger cells
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 4,607.67 grams
≈ 1.30 x 10⁶ cells

For smaller cells:
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 0.00000461 grams

≈ 1.30 x 10¹² cells
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⁶

The area of the shaded region is 5 sq units and the percentage of the shaded region is 83.33%

The area of the shaded region is the difference between the area of the rectangle and the area of the clear region

Assuming the following dimensions

So, we have

Shaded = 3 * 2 - 2 * 1/2 * 1 * 1

Evaluate

Shaded = 5

This is calculated as

Percentage = Shaded/Rectangle

So, we have

Percentage = 5/(3 * 2)

Evaluate

Percentage = 83.33%

Hence, the percentage of the shaded region is 83.33%

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Answer:

(2,-4)

Step-by-step explanation:

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

2x+3y=−8,3x+y=2

To make 2x and 3x equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 2. Then simplify

6x+9y=−24,6x+2y=4

Add 6x to −6x. Terms 6x and −6x cancel out, leaving an equation with only one variable that can be solved, add 9y to −2y, add −24 to −4, and divide both sides by 7.

y=−4

Substitute −4 for y in 3x+y=2. Because the resulting equation contains only one variable, you can solve for x directly. Add 4 to both sides of the equation and divide both sides by 3.

x=2

In linear equation, The sides of the triangle on the drawing should be 6 inches, 8 inches, and 9 inches.

What in mathematics is a linear equation?

An algebraic equation B. y=mx+b (where m is the slope and b is the y-intercept) containing simple constants and first-order (linear) components, such as the following, is called a linear equation.

                           The above is sometimes called a "linear equation in two variables" where x and y are variables. Equations in which the variable is power 1 are called linear equations. axe+b = 0 is a one-variable example where a and b are real numbers and x is a variable. 

A triangle with sides 12 feet, 16 feet, and 18 feet on a drawing where 1 inch = 2 feet.

Then, 1 feet of original triangle = 1/2 inch on drawing.

Now, the sides of the triangle on the drawing are

12 feet = 1/2 * 12 = 6 in

12 feet = 1/2 * 16 = 8 in

12 feet = 1/2 * 18 = 9 in

Hence, the sides of the triangle on the drawing should be 6 inches, 8 inches, and 9 inches..

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The four possible decodings of the ciphertext c = 823 845 737 are 156276219, 561472502, 1188260592, and 197895457.

To encode the plaintext m = 414 892 055, we first need to compute the corresponding ciphertext c using the Rabin cryptosystem.

The Rabin cryptosystem involves four steps: key generation, message encoding, message decoding, and key decryption. Since we already have the key n, we can skip the key generation step.

To encode the message m, we compute:

c ≡ m^2 (mod n)

Substituting the given values, we get:

c ≡ 414892055^2 (mod 1359692821)

c ≡ 1105307085 (mod 1359692821)

Therefore, the encoded ciphertext is c = 1105307085.

(b) To find the four decodings of the ciphertext c = 823 845 737, we need to use the Rabin cryptosystem to compute the four possible square roots of c modulo n.

First, we need to factorize n as n = 32359 · 42019. Then we compute the two square roots of c modulo each of the two prime factors, using the following formula:

x ≡ ± [tex]y^((p+1)/4) (mod p)[/tex]

where x is the square root of c modulo p, y is a solution to the congruence y^2 ≡ c (mod p), and p is one of the prime factors of n.

For the first prime factor p = 32359, we can use the following values:

y ≡ 3527^2 (mod 32359) ≡ 15467 (mod 32359)

x ≡ ± y^((p+1)/4) (mod p) ≡ ± 6692 (mod 32359)

Therefore, the two possible square roots of c modulo 32359 are 6692 and 25667.

For the second prime factor p = 42019, we can use the following values:

y ≡ 3527^2 (mod 42019) ≡ 25058 (mod 42019)

x ≡ ± y^((p+1)/4) (mod p) ≡ ± 1816 (mod 42019)

Therefore, the two possible square roots of c modulo 42019 are 1816 and 40203.

To find the four possible decodings of the ciphertext c = 823 845 737, we combine each of the two possible square roots modulo 32359 with each of the two possible square roots modulo 42019, using the Chinese Remainder Theorem:

x ≡ a (mod 32359)

x ≡ b (mod 42019)

where a and b are the two possible square roots modulo 32359 and 42019, respectively.

The four possible values of x are:

x ≡ 156276219 (mod 1359692821)

x ≡ 561472502 (mod 1359692821)

x ≡ 1188260592 (mod 1359692821)

x ≡ 197895457 (mod 1359692821)

Therefore, the four possible decodings of the ciphertext c = 823 845 737 are 156276219, 561472502, 1188260592, and 197895457.

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I think it might be C

The result of 19 minus a negative 29 is 48. Expressed as a fraction in simplest form, this would be 48/1.

To find the difference between 19 and negative 29, we can use the rule that subtracting a negative number is the same as adding its absolute value. So, 19 - (-29) is the same as 19 + 29, which equals 48.

To write this as a fraction in simplest form, we simply put 48 over 1, since any integer can be expressed as a fraction with a denominator of 1. We don't need to simplify any further, so our final answer is 48/1.

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The area of the stop sign, given the length of one side and the apothem, would be 308. 8 square inches.

To find the area of a regular octagon, we can use the formula:

Area = ( Perimeter × Apothem ) / 2

Initially, we must determine the perimeter of the octagon. Since it is a regular octagon all sides have an identical length. There are 8 sides with length equal to 8 inches; therefore the perimeter is:

= 8 x 8

= 64 inches

The area is;

Area = ( Perimeter × Apothem ) / 2

Area = ( 64 inches × 9.65 inches ) / 2

Area = 617. 6 square inches / 2

Area = 308.8 square inches

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[tex]SIGMA an = 1 + 2 + 3 + ... + n = n(n+1)/2 = ∫_1^n f(x)dx = ∫ f(x)dx[/tex]

f(x) = x is indeed a continuous function that satisfies the given condition.

Yes, we can find a continuous function f(x) such that when an = f(n), we have SIGMA an = ∫ f(x)dx.

One such function is f(x) = x.

To see why this works, let's consider a few terms of the series SIGMA an.

When n = 1, we have a1 = f(1) = 1, so the series starts with 1.

When n = 2, we have a2 = f(2) = 2, so the series becomes 1 + 2. When n = 3, we have a3 = f(3) = 3, so the series

becomes 1 + 2 + 3. And so on.

Notice that this series is just the sum of the first n positive integers, which we know is equal to n(n+1)/2.

But if we take the derivative of f(x) = x, we get f'(x) = 1, which means that the integral of f(x) from 1 to n is just n.

So we have:

[tex]∫ f(x)dx = ∫ xdx = 1/2 x^2 + C[/tex]

[tex]∫_1^n f(x)dx = (1/2 n^2 + C) - (1/2 (1)^2 + C) = 1/2 n^2 - 1/2[/tex]

And therefore:

[tex]SIGMA an = 1 + 2 + 3 + ... + n = n(n+1)/2 = ∫_1^n f(x)dx = ∫ f(x)dx[/tex]

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We are given that IQ scores are normally distributed with mean μ = 100 and standard deviation σ = 15. We want to find the probability of a random person on the street having an IQ score of less than 96.

To do this, we need to standardize the IQ score using the z-score formula:

z = (x - μ) / σ

where x is the IQ score we're interested in, μ is the mean IQ score, and σ is the standard deviation of IQ scores.

Plugging in the given values, we get:

z = (96 - 100) / 15 = -0.267

Now, we look up the probability of getting a z-score less than -0.267 in a standard normal distribution table or using a calculator. The probability is approximately 0.3944.

Therefore, the probability of a random person on the street having an IQ score of less than 96 is 0.3944 (rounded to 4 decimal places).

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The given statement "A function f(x) = 3x² dominates g(x) = x²" is True as it grows faster than the other function.

To show that f(x) dominates g(x), we need to prove that there exists a constant c such that f(x) > c * g(x) for all x > 0.

Let's consider c = 3. Then, for all x > 0, we have:

[tex]f(x) = 3x^2 > 3x^2/1 = 3x^2 * 1 > x^2 * 3 = g(x) * 3[/tex]

A function dominates another function when it grows faster than the other function. In this case, f(x) = 3x² and g(x) = x². Since f(x) has a higher coefficient (3) than g(x) (1) for the x² term, it grows faster than g(x) as x increases.

Therefore, we have shown that f(x) > 3g(x) for all x > 0, which means that f(x) dominates g(x).

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Hannah mixed 6.83 lb of pretzels with 3.57 lb of popcorn to make 10.4 lb of mixture. She then filled up 6 bags with an average of 1.68 lb of mixture per bag, leaving her with 0.35 lb of mixture left over.

In this problem, Hannah mixed 6.83 lb of pretzels with 3.57 lb of popcorn. This means that she had a total of 10.4 lb of mixture. She then filled up 6 bags that were the same size with the mixture, which means that each bag had approximately 1.73 lb of mixture (10.4 lb / 6 bags).

After filling up all 6 bags, Hannah had 0.35 lb of the mixture left over. This means that she used a total of 10.05 lb of mixture for the bags (10.4 lb - 0.35 lb).

To find out how much mixture was used per bag, we can divide the total amount of mixture used (10.05 lb) by the number of bags (6). This gives us an average of approximately 1.68 lb per bag.

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The equation that best represents the situation is y = 3/4x. The correct answer is A.

The given problem involves two people, Jamie and Anabel, who jogged a certain distance. Let's say Jamie jogged x km. Anabel jogged 1/4 less than Jamie, which means she jogged 3/4 of x km (since 1 - 1/4 = 3/4).

To represent this situation in an equation, we need to find the relationship between the distance jogged by Jamie and the distance jogged by Anabel. Since Anabel jogged 3/4 of the distance jogged by Jamie, we can write:

distance jogged by Anabel = 3/4(distance jogged by Jamie)

Using the given variable x for the distance jogged by Jamie, we can rewrite the equation as:

distance jogged by Anabel = 3/4x

And since the question is asking for an equation that best represents the situation, the correct answer is:

Cy = 3/4x

Therefore, Cy = 3/4x is the equation that best represents the situation where Jamie jogged x km and Anabel jogged 1/4 less than Jamie. The correct answer is A.

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Approximately 37.71 lbs of butter would be used to make 132 cookies.

we can use a proportion:

[tex]2 lbs of butter / 7 dozen cookies = x lbs of butter / 132 cookies\\[/tex]

To find x, we can cross-multiply and solve for x:

[tex]2 lbs of butter * 132 cookies = 7 dozen cookies * x lbs of butter264 lbs of cookies = 7xx = 264 / 7x = 37.71[/tex]

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We can start by integrating both sides of the differential equation to obtain:

∫f '(x) dx = ∫([tex]5x^2 - x^2/5[/tex]) dx

f(x) = (5/3)[tex]x^3[/tex] - (1/15) [tex]x^5[/tex] + C

where C is the constant of integration.

To find the value of C, we can use the initial condition f(1) = 0:

f(1) = (5/3)[tex](1)^3[/tex] - (1/15) [tex](1)^5[/tex] + C = 0

Simplifying this equation gives:

C = (1/15) - (5/3)

C = -2/9

Therefore, the solution to the initial value problem f '(x) = 5[tex]x^2[/tex] − [tex]x^2[/tex]/5 , f(1) = 0 is:

f(x) = (5/3) [tex]x^3[/tex] - (1/15) [tex]x^5[/tex] - (2/9)

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The height of the building is 16 meters.

What is right triangle?

A right triangle is a type of triangle that has one of its angles measuring 90 degrees (π/2 radians). The side which is opposite to the right angle is the hypotenuse, while the other two sides are called the legs.

We can solve this problem using the Pythagorean theorem, which relates the sides of a right triangle. Let h be the height of the building. Then we can draw a right triangle with one leg of length h and the other leg of length 12m, representing the height and length of the shadow, respectively. The hypotenuse of this triangle is the distance from the top of the building to the tip of the shadow, which is 20m. So we have:

h² + 12² = 20²

Simplifying and solving for h, we get:

h² = 20² - 12²

h² = 256

h = sqrt(256)

h = 16

Therefore, the height of the building is 16 meters.

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The image of point P (6, 1) under a 180° counterclockwise rotation about the origin is (-6, -1).

To find the image of point P (6, 1) under a 180° counterclockwise rotation about the origin, we can use the rotation formula for 2D coordinates.

The formula for rotating a point (x, y) counterclockwise by θ degrees about the origin is:

x' = x [tex]\times[/tex] cos(θ) - y [tex]\times[/tex] sin(θ)

y' = x [tex]\times[/tex] sin(θ) + y [tex]\times[/tex] cos(θ)

In this case, θ is 180°.

So, let's substitute the values of x and y from point P into the rotation formula:

x' = 6 [tex]\times[/tex] cos(180°) - 1 [tex]\times[/tex] sin(180°)

y' = 6 [tex]\times[/tex] sin(180°) + 1 [tex]\times[/tex] cos(180°)

Now, let's simplify these equations using the trigonometric values for 180°:

[tex]x' = 6 \times (-1) - 1 \times 0[/tex]

[tex]y' = 6 \times 0 + 1 \times (-1)[/tex]

Simplifying further:

x' = -6

y' = -1

Therefore, the image of point P (6, 1) under a 180° counterclockwise rotation about the origin is (-6, -1).

Please note that the rotation formula assumes angles are measured in radians.

However, for simplicity, we used degrees in this explanation.

The trigonometric functions (cos and sin) can be evaluated in radians using their corresponding values for 180°.

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