8. A square has a side length of 11 V2 meters. What is the length of the diagonal
of the square?

Answer:

350/80

cancel out the zeros

35/8

4 3/8 or 4.375

350 divided by 80 is equal to 4 with a remainder of 30, or in decimal form, it is approximately 4.375.

We have,

To divide 350 by 80, we can perform the division operation as follows:

- First, we check how many times 80 can be divided into 350. We start with the largest multiple of 80 which is less than or equal to 350, which is 4.

80 x 4 = 320

- We subtract 320 from 350 to find the remainder:

350 - 320 = 30

- Since the remainder is not zero,

We can continue dividing. We bring down the next digit of 350, which is 0.

- Now, we have 300 as the new dividend.

We ask ourselves how many times 80 can be divided into 300.

80 x 3 = 240

- Subtracting 240 from 300 gives us the new remainder:

300 - 240 = 60

- Again, the remainder is not zero, so we continue.

- We bring down the last digit of 350, which is 0, and our new dividend becomes 600.

- We ask ourselves how many times 80 can be divided into 600.

80 x 7 = 560

- Subtracting 560 from 600 gives us the new remainder:

600 - 560 = 40

- The remainder is still not zero, so we continue.

- Finally, we bring down the last digit of 350, which is 0.

Our new dividend is 400.

We ask ourselves how many times 80 can be divided into 400.

80 x 5 = 400

- Subtracting 400 from 400 gives us zero as the remainder.

Since the remainder is now zero, we can stop dividing.

Therefore,

350 divided by 80 is equal to 4 with a remainder of 30, or in decimal form, it is approximately 4.375.

In summary, 350 divided by 80 equals 4 with a remainder of 30, or 4.375.

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Properties B and C together ensure that there exists a 'c' in (-1, 1) at which f'(c) = 0.
Your answer: 1. B and C only

Which properties ensure that there exists ?

a 'c' in (-1, 1) at which f'(c) = 0, given f(-1) = f(1) = and the properties A, B, and C.

First, let's define the properties:
f(1) = 2
f is continuous on (-1, 1]
f(x) = (22/3) - x^2

To ensure that there exists a 'c' in (-1, 1) at which f'(c) = 0, we will use the Mean Value Theorem (MVT). MVT states that if a function is continuous on a closed interval and differentiable on an open interval, then there exists at least one 'c' in the interval where the derivative is 0.

Looking at property B, it states that f is continuous on (-1, 1], which satisfies the first condition of the MVT. Property C provides a specific function for f(x), which is differentiable on (-1, 1) since it is a polynomial function. Therefore, properties B and C together ensure that there exists a 'c' in (-1, 1) at which f'(c) = 0.

Your answer: 1. B and C only

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Answer:

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

The total cost of repair is expressed by the function:

As we see,

Therefore the answer is option A.

The amount it will cost to buy enough stain for the whole porch is $182.

To determine the cost of staining Wanda's front porch, we'll first need to calculate the area of the porch, which is a rectangle in shape. The area of a rectangle can be found using the formula: Area = length × width.

In this case, the length of Wanda's porch is 13 feet, and the width is 7 feet. So, we'll multiply these two measurements to find the area:

Area = 13 ft × 7 ft = 91 square feet

Now that we know the area, we can calculate the cost of the stain. The stain costs $2.00 per square foot, so we'll multiply the area of the porch (91 square feet) by the cost per square foot:

Cost = 91 sq ft × $2.00/sq ft = $182

Therefore, it will cost Wanda $182 to buy enough stain for the whole porch.

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$148.39 much money will he have to spend for his skateboard.

Levi will have to spend approximately $148.39 for his skateboard.

The original price can be defined as the cost price of an item or a service. The decrease in the original price of a product or service is called the discount offered to the buyer. Generally, this discount is expressed as a percentage.

Original Sale Price means the price at which the current Owner purchased the Property (not including commissions, loan origination fees, appraisals fees, title insurance premiums and other similar transaction costs).

To calculate this, we need to find 6% of the original price and add it to the original price:

6% of 139.99 = 0.06 x 139.99 = 8.3994

Adding this to the original price gives:

139.99 + 8.3994 = 148.3894

Rounding to the nearest cent gives $148.39.

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The percentage of $7,200 that was donated to charity would be = 2%

To calculate the percentage of the total sales that was donated, the following should be carried out.

The total sales at the shoe store = $7,200

The amount of money that was donated = $144

Therefore to calculate the percentage the following is done ;

= 144/7200 × 100/1

= 14400/7200

= 2%

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The expression that represents the amount of ore sold and how much ore can the mine sell after extracting ore for 12 hours is option B: The expression is 2t−14t. The amount of ore is 21 metric tons.

The reasoning for the selection of the expression and amount of ore can the mine sell after extracting ore for 12 hours  is as follows.

1: Determine the amount of coal used for electricity generation in terms of t.

The mine uses 14 tons of coal every hour, so the total amount used for electricity generation is 14t.

2: Determine the total amount of coal extracted in terms of t.

The mine extracts 2 tons of coal every hour, so the total amount extracted is 2t.

3: Calculate the amount of coal sold in terms of t.

To find the amount of coal sold, subtract the amount used for electricity generation from the total amount extracted: 2t - 14t.

4: Determine the amount of coal sold after 12 hours.

Substitute t = 12 into the expression:

2(12) - 14(12) = 24 - 168 = -144.

However, since the mine uses 14 tons of the extracted coal every hour, it cannot sell more coal than it extracts. So, the correct expression should be 2t - 14 (without the t for the amount used for electricity generation).

5: Calculate the amount of coal sold after 12 hours using the corrected expression.

Substitute t = 12 into the expression: 2(12) - 14 = 24 - 14 = 10 metric tons.

The correct expression should be 2t - 14, and the amount of coal the mine can sell after extracting coal for 12 hours is 10 metric tons. Hence, the correct answer is option B.

Note: The question is incomplete. The complete question probably is: A mine extracts 2 metric tons of coal in an hour. The mine uses 14 ton of the extracted coal every hour to generate electricity for the mine and sells the rest. If t is the number of hours spent mining, which expression represents the amount of ore sold? How much ore can the mine sell after extracting ore for 12 hours? A) The expression is 2t−1/4t. The amount of ore is 23 3/4 metric tons. B) The expression is 2t−1/4t. The amount of ore is 21 metric tons. C) The expression is 2t+1/4t. The amount of ore is 24 metric tons. D) The expression is 2t+1/4t. The amount of ore is 24 1/4 metric tons.

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To prove that the right triangles are congruent, we need to show that they have three pairs of congruent parts (sides or angles).

Let's say that the given congruent parts are the hypotenuses and one leg of each triangle. To prove congruence, we can add one more pair of congruent parts, such as the other leg.

By the Side-Angle-Side (SAS) congruence theorem, if two triangles have two pairs of congruent sides and the included angle is also congruent, then the triangles are congruent. In this case, we have two pairs of congruent sides (the hypotenuses and one leg) and the included angle (the right angle) is congruent by definition.

Therefore, we can conclude that the two right triangles are congruent by SAS.

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To find the surface area of a spherical object, we need to know the radius of the sphere. However, in this case, only the circumference of the pincushion is given, which is not enough information to directly determine the radius.

The formula relating the circumference (c) and the radius (r) of a sphere is:

c = 2πr

To find the surface area (A) of the sphere, we can use the formula:

A = 4πr^2

Since we don't have the radius, we need to solve the circumference formula for the radius first:

c = 2πr

Divide both sides of the equation by 2π:

r = c / (2π)

Now we can substitute the value of c = 18 cm into the equation to find the radius:

r = 18 cm / (2π)

r ≈ 2.868 cm (approximately)

Now that we have the radius, we can calculate the surface area using the formula:

A = 4πr^2

A = 4π(2.868 cm)^2

A ≈ 103.05 cm² (approximately)

Therefore, the surface area of the pincushion is approximately 103.05 square centimeters.

Using FOIL, the simplified expression for the product of (3x-4)(6x-2) is c. 18x² - 30x + 8.

To simplify the product (3x-4)(6x-2) using FOIL, we follow the First, Outer, Inner, Last rule. Let's break down the process:

First: Multiply the first terms of both expressions:
(3x) * (6x) = 18x²

Outer: Multiply the outer terms of both expressions:
(3x) * (-2) = -6x

Inner: Multiply the inner terms of both expressions:
(-4) * (6x) = -24x

Last: Multiply the last terms of both expressions:
(-4) * (-2) = 8

Now, combine the results:
18x² - 6x - 24x + 8

Simplify by combining the like terms (middle terms -6x and -24x):
18x² - 30x + 8

The simplified product is 18x² - 30x + 8, which corresponds to option (c).

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Since f''(x) is always 0, f(x) is not concave upward on any interval.

On what interval is f concave upward?

The first derivative of f(x) is f'(x) = 6.
The second derivative of f(x) is f''(x) = 0.
The interval on which f(x) is increasing is when f'(x) > 0, which is when x is in the interval (-2,-5) U (1,60).
The interval on which f(x) is decreasing is when f'(x) < 0, which is when x is in the interval (-5,1).
The interval on which f(x) is concave downward is when f''(x) < 0, which is all values of x.
The interval on which f(x) is concave upward is when f''(x) > 0, which is no values of x.

To find the first derivative of f(x), we need to take the derivative of each term separately. The derivative of 3 is 0, the derivative of 6x is 6, and the derivative of -153 +3.2 is 0. Adding these up gives us f'(x) = 6.

To find the second derivative of f(x), we need to take the derivative of f'(x), which is a constant function. The derivative of a constant function is always 0, so f''(x) = 0.

To determine where f(x) is increasing, we need to find the values of x where f'(x) > 0. Since f'(x) is a constant function, it is always positive, so f(x) is increasing on the interval (-2,-5) U (1,60).

To determine where f(x) is decreasing, we need to find the values of x where f'(x) < 0. Since f'(x) is a constant function, it is always positive, so f(x) is decreasing on the interval (-5,1).

To determine where f(x) is concave downward, we need to find the values of x where f''(x) < 0. Since f''(x) is always 0, f(x) is concave downward on all values of x.

To determine where f(x) is concave upward, we need to find the values of x where f''(x) > 0. Since f''(x) is always 0, f(x) is not concave upward on any interval.

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a. The actual area of the kitchen is 128 square feet.

b.  the couch measures 1.25 inches across in the scale drawing.

The scale is 1/4 inch = 2 feet,

meaning that in the drawing, each inch represents 8 feet (since 2 feet/0.25 inches = 8 feet/inch).

The kitchen in the drawing has an area of 2 square inches.

we apply  the scale factor to find the actual area in square feet,

1 inch in the drawing = 8 feet in real life

So, 2 square inches in the drawing = 2 x 8 x 8 = 128 square feet in real life.

Hence,  the actual area of the kitchen is 128 square feet.

we also apply  the scale factor to  find the couch measurement in the scale drawing,

1 inch in the drawing = 8 feet in real life

10 feet in real life = 10/8 = 1.25 inches in the drawing.

In conclusion,  the couch measures 1.25 inches across in the scale drawing.

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To determine whether a critical point is a relative minimum or maximum using concavity, we need to examine the second derivative of the function at the critical point.

If the second derivative is positive, then the function is concave up, meaning it is shaped like a bowl opening upwards. At a critical point where the first derivative is zero, this indicates a relative minimum, as the function is increasing on either side of the critical point.

On the other hand, if the second derivative is negative, then the function is concave down, meaning it is shaped like a bowl opening downwards. At a critical point where the first derivative is zero, this indicates a relative maximum, as the function is decreasing on either side of the critical point.

If the second derivative is zero, then the test is inconclusive and further analysis is needed, such as examining higher order derivatives or using other methods such as the first derivative test.

Therefore, the concavity test is a useful method for determining the nature of critical points and whether they represent a relative minimum or maximum.

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Complete question is:

What we need to examine for a method for determining whether a critical point is a relative minimum or maximum using concavity.

68% of the stores will sell a gallon of milk for between $3.57 and $3.93 with one standard deviation of the mean.

Mean =  $3.75

Standard deviation = $0.09.

Selling variations = $3.57 to $3.93.

Chebyshev's theorem states that if k is a positive number, then, in any data at least [tex](1 - 1/k^2)[/tex] of the data will fall in K standard deviations from the mean data.

For normal distributions, 68% of the values will fall within one standard deviation of the mean.

For non-normal deviations, 75% of values will fall within 2 standard deviations of the mean.

Here we need to find the Upper limit and lower limit of the data.

$3.75 - $0.09 = $3.66

$3.75 + $0.09 = $3.84

The price range will be between $3.66 and $3.84.

To calculate the minimum percentage of stores using Chebyshev's theorem

minimum percentage = [tex]1 - 1/k^2[/tex]

minimum percentage = [tex]1 - 1/(1^2)[/tex]

minimum percentage = 0

Here, 0% of stores will fall with only one standard deviation from the Mean.

Therefore, we can conclude that 68% of the stores will sell a gallon of milk for between $3.57 and $3.93.

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The population grows by a factor of 2 every year.

The number of fish in a lake is growing exponentially, with given data for different numbers of years since the population was first measured as follows:

Years | Population (in thousands)
0     | 10
1     |
2     | 40
3     |
4     |
5     |
6     |

By what factor does the population grow every two years?
From year 0 to year 2, the population grows from 10,000 to 40,000. To find the growth factor, divide the population in year 2 by the population in year 0:
40,000 ÷ 10,000 = 4.

Thus, the population grows by a factor of 4 every two years.

Using this information, we can calculate the population for years 4 and 6:
Year 4: 40,000 × 4 = 160,000 (160 in thousands)
Year 6: 160,000 × 4 = 640,000 (640 in thousands)

By what factor does the population grow every year?
To find the yearly growth factor, take the square root of the growth factor every two years:
√4 = 2.

Thus, the population grows by a factor of 2 every year.

Using this information, we can complete the table:
Years | Population (in thousands)
0     | 10
1     | 20
2     | 40
3     | 80
4     | 160
5     | 320
6     | 640

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The cost of apples and bananas are $ 0.70 and $ 1.10 respectively if Patricia bought 4 apples and 9 bananas for $12.70 and Jose got 8 apples and 11 bananas for $17.70

Let the cost of one apple be a

the cost of one banana be b

In the case of Patricia,

12.70 = cost of 4 apples + cost of 9 bananas

Cost of 4 apples = 4a

Cost of 9 bananas = 9b

The equation we get is

4a + 9b = 12.70 ----(i)

In the case of Jose,

17.70 = cost of 8 apples + cost of 11 bananas

Cost of 8 apples = 8a

Cost of 11 bananas = 11b

The equation we get is

8a + 11b = 17.70 ----(ii)

Multiply (i) by 2

8a + 18b = 25.40 --- (iii)

Subtract (ii) and (iii)

7b = 7.70

b = $ 1.10

4a + 9 (1.10) = 12.70

4a + 9.90 = 12.70

4a = 2.80

a = $ 0.70

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Step-by-step explanation:

[tex] \frac{1}{( {x}^{2} + 1) } = \frac{u}{v} [/tex]

u = 1

u' = 0

v = x² + 1

v' = 2x

[tex] \frac{1}{ ({x}^{2} + 1)} \\ = \frac{u'v - v'u}{ {v}^{2} } \\ = \frac{0 - (2x \times 1)}{ {( {x}^{2} + 1)}^{2} } \\ = - \frac{2x}{ { ({x}^{2} + 1) }^{2} } [/tex]

The volume of the cylinder after the rotation of rectangle is V = 12π units³ = 37.68 units³

Given data ,

Let the width of the rectangle be w = 2 units

Let the length of the rectangle be l = 3 units

Now , on rotating a rectangle , we get a cylinder

And , the radius of the cylinder = 2 units

And , the height of the cylinder = 3 units

Now , Volume of Cylinder = πr²h

On simplifying , we get

V = π ( 2 )² ( 3 )

V = 12π units³

V = 37.68 units³

Hence , the volume of cylinder is V = 37.68 units³

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Using the seven square tiles numbered 2, 3, 4, 5, 6, 7, and 8, we can make 1260 different 4-digit numbers.

To create a 4-digit number using these seven square tiles, we have to consider the following:

- The first digit cannot be 2 because then the number would only have three digits.
- We can choose any of the remaining six tiles for the first digit, which means there are 6 choices.
- We can choose any of the seven tiles for the second digit, which means there are 7 choices.
- We can choose any of the remaining six tiles for the third digit, which means there are 6 choices.
- We can choose any of the remaining five tiles for the fourth digit, which means there are 5 choices.

Therefore, the total number of 4-digit numbers we can make is:

6 x 7 x 6 x 5 = 1260

So, using the seven square tiles numbered 2, 3, 4, 5, 6, 7, and 8, we can make 1260 different 4-digit numbers.

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The probability that there will be at least 4 typos on page 301 is 0.847

To solve this problem, we need to make some assumptions. Let's assume that the number of typos on each page follows a Poisson distribution with a mean of 6 typos per page, and that the number of typos on one page is independent of the number of typos on any other page.

Under these assumptions, we can use the Poisson distribution to calculate the probability of observing a certain number of typos on a given page.

Let X be the number of typos on page 301. Then X follows a Poisson distribution with a mean of 6 typos per page. The probability of observing at least 4 typos on page 301 can be calculated as follows

P(X ≥ 4) = 1 - P(X < 4)

= 1 - P(X = 0) - P(X = 1) - P(X = 2) - P(X = 3)

Using the Poisson distribution formula, we can calculate the probabilities of each of these events

P(X = k) = (e^-λ × λ^k) / k!

where λ = 6 and k is the number of typos. Thus,

P(X = 0) = (e^-6 × 6^0) / 0! = e^-6 ≈ 0.0025

P(X = 1) = (e^-6 × 6^1) / 1! = 6e^-6 ≈ 0.015

P(X = 2) = (e^-6 × 6^2) / 2! = 18e^-6 ≈ 0.045

P(X = 3) = (e^-6 × 6^3) / 3! = 36e^-6 ≈ 0.091

Plugging these values into the equation above, we get

P(X ≥ 4) = 1 - (e^-6 + 6e^-6 + 18e^-6 + 36e^-6)

≈ 0.847

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To multiply the polynomials (1-2t)(5t+t^2), we can use the distributive property and multiply each term in the first polynomial by each term in the second polynomial:

(1-2t)(5t+t^2) = 1(5t+t^2) - 2t(5t+t^2)

Multiplying the first term by each term in the second polynomial, we get:

5t + t^2

Multiplying the second term by each term in the second polynomial, we get:

-10t^2 - 2t^3

Combining like terms, we get:

-2t^3 - 10t^2 + 5t

Therefore, the answer is D. -2t^3 - 9t^2 + 5t.

The probability that a given voicemail is between 20 and 50 seconds is 0.8185, or 81.85%.

To find the probability that a voicemail is between 20 and 50 seconds, we need to standardize the values and use a standard normal distribution table.

First, we find the z-scores for 20 seconds and 50 seconds:

z1 = (20 - 40) / 10 = -2

z2 = (50 - 40) / 10 = 1

Using a standard normal distribution table, we can find the area to the left of each z-score:

Area to the left of z1 = 0.0228

Area to the left of z2 = 0.8413

To find the probability between 20 and 50 seconds, we subtract the area to the left of z1 from the area to the left of z2:

P(20 < x < 50) = P(-2 < z < 1)

= 0.8413 - 0.0228

= 0.8185

Therefore, the probability that a given voicemail is between 20 and 50 seconds is 0.8185, or 81.85%.

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The height of the cone is approximately 3.4 cm.

The formula for the volume of a cone is V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.

We are given that the radius of the cone is 3 cm and the volume is 94.2 cm^3. Substituting these values into the formula, we get:

94.2 = (1/3)π(3^2)h

Simplifying:

94.2 = 9πh

h = 94.2 / (9π)

h ≈ 3.4

Therefore, the height of the cone is approximately 3.4 cm.

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Using Newton's method, the root of the equation cos(x) = x correct to six decimal places is approximately 0.739085.

To solve the equation cos(x) = x, we can use Newton's method, which involves repeatedly applying an iterative formula to approximate the root of the equation. We first rewrite the equation in the form f(x) = cos(x) - x = 0 and find its derivative f'(x) = -sin(x) - 1.

The iterative formula for Newton's method is given by xn+1 = xn - f(xn)/f'(xn). Applying this formula, we get xn+1 = xn + cos(xn) - xn sin(xn) + 1.

To start the iteration, we need to guess a suitable value for x1. From the graph of y = cos(x) and y = x, we can see that they intersect at a point whose x-coordinate is slightly less than 1. Therefore, we take x1 = 1 as a convenient first approximation.

Using a calculator in radian mode, we can apply the iterative formula to obtain the following approximations:

x2 ≈ 0.75036387

x3 ≈ 0.73911289

x4 ≈ 0.73908513

x5 ≈ 0.73908513

Since x4 and x5 agree to six decimal places, we can conclude that the root of the equation cos(x) = x correct to six decimal places is approximately 0.739085.

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The simplified rational expression for this problem is given as follows:

[tex]\frac{3\sqrt{21} + 13}{12}[/tex]

The rational expression in the context of this problem is defined as follows:

[tex]\frac{\sqrt{7} + \sqrt{3}}{2\sqrt{3} - \sqrt{7}}[/tex]

The first step in simplifying the expression is removing the root from the denominator, multiplying numerator and denominator by the conjugate, as follows:

[tex]\frac{\sqrt{7} + \sqrt{3}}{2\sqrt{3} - \sqrt{7}} \times \frac{2\sqrt{3} + \sqrt{7}}{2\sqrt{3} + \sqrt{7}}[/tex]

Applying the subtraction of perfect squares, the denominator is given as follows:

2² x 3 - 7 = 12.

The numerator is:

[tex](\sqrt{7} + \sqrt{3})(2\sqrt{3} + \sqrt{7}) = 2\sqrt{21} + 7 + 6 + \sqrt{21} = 3\sqrt{21} + 13[/tex]

Thus the simplified expression is:

[tex]\frac{3\sqrt{21} + 13}{12}[/tex]

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The area of the lawn that is watered by the sprinkler is approximately 200.96 square feet.

The area of the field that's  doused  by the sprinkler is a sector of a circle with a compass of 8  bases and a central angle of 360 degrees.

To find the area of the sector, we can use the formula

Area = ( θ/ 360) x πr2

 where θ is the central angle in degrees,

r is the radius of the circle,

and π is the constant pi.  

Substituting the given values,

we get   Area = (360/360) x3.14 x 82

Area = 3.14 x 64  Area = 200.96

Thus, the area of the field that's  doused  by the sprinkler is  roughly 200.96 square feet.

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The complete question is as follows:

Michael is using a rotating sprinkler to water his lawn. The sprinkler rotates in a complete circle. It sprays water at most 8 feet. Find the area of the lawn that is watered.

The perimeter of the larger kennel will be twice as large as the perimeter of the original kennel.

To find the ratio of the perimeters of the original kennel to the larger kennel, we need to know how doubling the dimensions affects the perimeter.

Since the perimeter is the sum of all four sides, doubling each side will result in a perimeter that is double the original.

Therefore, the perimeter of the larger kennel will be two times the perimeter of the original kennel.

In mathematical terms:

Perimeter of larger kennel = 2 x perimeter of original kennel

So, the perimeter of the larger kennel will be twice as large as the perimeter of the original kennel.

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The measure of angle x is: x = (3/5)(180 - z) = (3/5)(180 - 90) = 36 degrees

And the measure of angle y is:y = (2/5)(180 - z) = (2/5)(180 - 90) = 24 degrees

Since lines lll, mmm, and nnn are parallel to each other and ppp is a transversal, we can use the angle properties of parallel lines to find the relationship between angle x and angle y.

From the given information, we have:  2x = 3y

Simplifying this equation, we get:  x = (3/2)y

Now, we can use this relationship to find the measures of angles x and y in terms of a common variable. Let's use z as the common variable.

x + y + z = 180 (angles on a straight line)

Substituting x = (3/2)y, we get:  (3/2)y + y + z = 180

Simplifying this equation, we get: (5/2)y + z = 180

Now, we can express y in terms of z:

(5/2)y = 180 - z

y = (2/5)(180 - z)

Similarly, we can express x in terms of z:

x = (3/2)y = (3/2)(2/5)(180 - z) = (3/5)(180 - z)

Now, we can use the relationship between angle x and angle y to find the measure of angle x in terms of z: 2x = 3y

2[(3/5)(180 - z)] = 3[(2/5)(180 - z)]

Simplifying this equation, we get: (6/5)z = 108

z = (5/6)(108) = 90

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