Hello, please help me with this geometry question asap. (The question is in the image below) thank you!

To put the numbers in order from least to greatest, we can use the number line: -3.5 -2.1 -1 -0.5 0.5 2 4 4.2 5 5.8  Two numbers that are opposites and each more than 6 units away from 0 are -7 and 7.

First, let's put the numbers in order from least to greatest:

-3.5, -2.1, -1.5, -1, -0.5, 0.5, 3.5, 4, 4.2, 4.8, 5

Now, let's find two numbers that are opposites and each more than 6 units away from 0. One example would be -7 and 7. These numbers are opposites (since they have the same magnitude but different signs), and they are both more than 6 units away from 0.

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The volume of the pyramid with a square base of side length 16.6 meters and a height of 9.1 meters is approximately 836.6 cubic meters.

To find the volume of a pyramid with a square base, you'll need to know the side length of the base and the height of the pyramid. In this case, the side length of the square base is 16.6 meters, and the height of the pyramid is 9.1 meters. Here's a step-by-step explanation to calculate the volume:

1. Find the area of the square base: Since the base is a square, you'll need to multiply the side length by itself.
Area = side_length × side_length
Area = 16.6 m × 16.6 m
Area ≈ 275.56 m²

2. Calculate the volume of the pyramid: To find the volume, you'll multiply the area of the base by the height of the pyramid and divide the result by 3.
Volume = (Area × Height) / 3
Volume ≈ (275.56 m² × 9.1 m) / 3
Volume ≈ 836.626 m³

3. Round the answer to the nearest tenth of a cubic meter:
Volume ≈ 836.6 m³

So, the volume of the pyramid with a square base of side length 16.6 meters and a height of 9.1 meters is approximately 836.6 cubic meters.

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To withdraw $3,200 quarterly at an interest rate of 4.5% for 18 years, the account balance needs to be approximately $178,311. This is calculated using the formula for the present value of an annuity, where the payment, interest rate, time period, and compounding frequency are considered.

To find the account balance needed, we need to use the present value of an annuity formula.

Convert the annual interest rate to a quarterly rate: 4.5% / 4 = 1.125%

Convert the number of years to the number of quarters: 18 years * 4 quarters per year = 72 quarters

Calculate the present value of the annuity using the formula:

PV = PMT * (1 - (1 + r)⁻ⁿ) / r

where PV is the present value, PMT is the regular withdrawal amount, r is the quarterly interest rate, and n is the number of quarters.

Plugging in the values, we get

PV = 3200 * (1 - (1 + 0.01125)⁻⁷²) / 0.01125

= 3200 * (1 - 0.2717) / 0.01125

= 178,311.11

Round the answer to the nearest dollar: $178,311

Therefore, the account needs to hold $178,311 to make regular withdrawals of $3200 per quarter for 18 years at a quarterly interest rate of 4.5%.

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Both situations can be represented by the number line as they both involve values either greater than or less than 75.

The number line the teacher drew can represent both Brooke's and Eileen's situations.

In Brooke's situation, the number line can represent the amount of money Christopher needs to earn to buy a new bicycle. If he needs to earn more than $75, any point on the number line greater than 75 would represent the amount of money he has earned that is sufficient for purchasing the bicycle.

In Eileen's situation, the number line can represent the time left in Paul's flight. If Paul has less than 75 minutes left, any point on the number line less than 75 would represent the time remaining in his flight.

Both situations can be represented by the number line as they both involve values either greater than or less than 75.

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Valterri would finish 1.1776 minutes (or 70.656 seconds) faster than if he raced at his Day 1 rate, if he raced at his Day 2 rate for the entire 64-lap race

To calculate how much faster Valterri would finish the full race if he raced at his Day 2 rate compared to his Day 1 rate, we need to first calculate his time difference per lap.

On Day 1, Valterri's rate was 3.2, which means he completed each lap in 1/3.2 or 0.3125 minutes (18.75 seconds).

On Day 2, his rate was 3.4, so he completed each lap in 1/3.4 or 0.2941 minutes (17.65 seconds).

The time difference per lap between Day 1 and Day 2 is 0.3125 - 0.2941 = 0.0184 minutes (or 1.104 seconds).

To find out how much faster Valterri would finish the full race if he raced at his Day 2 rate, we need to multiply this time difference per lap by the number of laps in the race.

The race has 64 laps, so:

Time difference = 0.0184 x 64 = 1.1776 minutes (or 70.656 seconds)

Therefore, if Valterri raced at his Day 2 rate for the entire 64-lap race, he would finish 1.1776 minutes (or 70.656 seconds) faster than if he raced at his Day 1 rate.

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The error in the calculation is that n/360 should be equal to the central angle of the sector in degrees divided by 360. However, the given value of 115/225 is not the correct central angle. To find the correct central angle, we need to use the formula for the area of a sector:

Area of sector XZY = (central angle/360) x πr^2

We know that the area of circle ⊙Z is 255 square feet, so we can find the radius:

πr^2 = 255
r^2 = 81.11
r ≈ 9 feet

Now we can solve for the central angle:

Area of sector XZY = (central angle/360) x π(9)^2
Area of sector XZY = (central angle/360) x 81π
Area of sector XZY = (central angle/360) x 254.47

Since the area of sector XZY is not given, we cannot use the given equation n/360 = 115/225 to find the central angle. Instead, we need to use the formula above and solve for the central angle. Let A be the area of sector XZY:

A = (n/360) x 254.47
n/360 = A/254.47
n = 360A/254.47

Now we can substitute the given area of circle ⊙Z and solve for the area of sector XZY:

255 = (n/360) x πr^2
255 = (n/360) x π(81)
255 = (n/360) x 254.47
n = (360 x 255)/254.47
n ≈ 360.15

Note that n should be rounded to the nearest integer since it represents the central angle in degrees. Therefore, the central angle is approximately 360 degrees. Now we can use this value to find the area of sector XZY:

Area of sector XZY = (360/360) x π(9)^2
Area of sector XZY = 81π
Area of sector XZY ≈ 254.47 ft^2

Therefore, the area of sector XZY should be approximately 254.47 square feet.

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The value of the variables are;

x = 52

y = 30

from the information given, we have simultaneous equations ;

−7y−4x=1

7y−2x=53

Make 'y' the subject from equation 1 , we have;

y = 1 + 4x/-7

Substitute the value into equation 2, we get;

7(1 + 4x/-7) - 2x = 53

expand the bracket

7 + 28x/-7 - 2x= 53

7 + 28x + 14x = 53(-7)

then, we have;

7 + 42x =,-371

collect the like terms

42x = 364

x = 52

Substitute the value

y =  1 + 4x/-7

y = 1+ 4(52)/-7

y = 30

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Answer:

B

Step-by-step explanation:

using the diamond factoring method:

2x^2-12x+x-6

2x(x-6) + (x-6)

(2x+1)(x-6)

B

The approximate total amount of string needed to keep the tree perpendicular to the ground is 4.02 feet, which is closest to answer choice C, 5.83 ft.

Assuming that Luis attached the ropes at the same height on the tree, the length of the rope needed for each side of the tree would be equal to the distance from the tree to the stake.

To keep the tree perpendicular to the ground, the distance from the tree to the stake should be equal to half of the diameter of the tree's canopy.

However, since the diameter of the canopy is not given, we can estimate it based on the height of the tree.

According to some tree experts, the average height-to-canopy-diameter ratio for a mature tree is about 5:1.

This means that if the tree is 20 feet tall, its canopy diameter is approximately 4 feet.

Using this estimate, we can assume that the canopy diameter of Luis's tree is about 4 feet, or 1.33 yards.

Thus, the distance from the tree to the stake should be approximately 0.67 yards.

Since there are two sides of the tree, Luis would need a total of 2 times 0.67 yards, or approximately 1.34 yards of rope.

Converting yards to feet, we get:

[tex]1.34 yards * 3 feet/yard = 4.02 feet[/tex]

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

Given that, the area of a hexagon is about 65 square units. You decompose the figure into 6 triangles.

A regular hexagon can be decomposed into 6 equal triangles,

So, the area of each triangle is 65/6 = 10.8 square units

The height of one triangle is about 4.3 units.

We know that, the area of a triangle is 1/2 ×Base×Hieght

Now, 10.8=1/2 ×Base×4.3

21.6=Base×4.3

Base=21.6/4.3

Base=5.02

Therefore, the approximate length of the base of the triangle is 5 units.

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Answer: 9.33

Step-by-step explanation: if you add them up, it's 9.33

The bank needs to buy and sell 20,000 dollars.

To calculate the amount of American dollars that should be bought and sold by the bank to earn a profit of Rs 9000, we first need to determine the exchange rate difference between the buying and selling rates:

Exchange rate difference = selling rate - buying rate

Exchange rate difference = Rs 117.30 - Rs 116.85

Exchange rate difference = Rs 0.45

This means that for every dollar bought and sold by the bank, there is a difference of Rs 0.45. To earn a profit of Rs 9000, we need to find out how many dollars the bank needs to buy and sell to make this amount of profit.

Let X be the amount of American dollars the bank needs to buy and sell to earn a profit of Rs 9000.

Profit = Exchange rate difference × X

Rs 9000 = Rs 0.45 × X

To calculate the amount of American dollars that should be bought and sold by the bank to earn a profit of Rs 9000, we first need to determine the exchange rate difference between the buying and selling rates:

Exchange rate difference = selling rate - buying rate

Exchange rate difference = Rs 117.30 - Rs 116.85

Exchange rate difference = Rs 0.45

This means that for every dollar bought and sold by the bank, there is a difference of Rs 0.45. To earn a profit of Rs 9000, we need to find out how many dollars the bank needs to buy and sell to make this amount of profit.

Let X be the amount of American dollars the bank needs to buy and sell to earn a profit of Rs 9000.

Profit = Exchange rate difference × X

Rs 9000 = Rs 0.45 × X

To solve for X, we can divide both sides by 0.45:

X = Rs 9000 ÷ Rs 0.45

X = 20,000

Therefore, the bank needs to buy and sell 20,000 American dollars to earn a profit of Rs 9000.

To calculate the amount of American dollars the bank needs to buy and sell, we first need to determine the exchange rate difference between the buying and selling rates. This is done by subtracting the buying rate from the selling rate. The resulting exchange rate difference gives us the profit the bank earns for every dollar bought and sold.

Next, we use the exchange rate difference to calculate the amount of American dollars needed to earn a profit of Rs 9000. We set up an equation where the profit is equal to the exchange rate difference multiplied by the amount of American dollars bought and sold. We solve for X, which represents the amount of American dollars needed to earn the profit of Rs [tex]9000.[/tex]

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To find the probability P(X < 4) for the given probability model, where X represents the number of cups of coffee an adult aged 18 or over drinks on average per day in the United States. The probabilities for each number of cups are given in the table:
- 2 cups: 0.36
- 3 cups: 0.19
- 4 or more cups: 0.11

To find P(X < 4), we need to sum the probabilities of X being 2 or 3 cups, as those are the only values less than 4:

P(X < 4) = P(X = 2) + P(X = 3)
P(X < 4) = 0.36 + 0.19

Now, we just need to add these probabilities together:

P(X < 4) = 0.55

So, the probability that a randomly chosen adult drinks fewer than 4 cups of coffee per day is 0.55 or 55% when expressed as a percentage.

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The value of the trigonometric ratio tanA from the right angle triangle is 3/4.

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled

To find the value of the trigonometric ratio tanA from the right angle triangle, we use the formula below

Formula:

From the right angle triangle,

Substitute these values into equation 1

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To determine how many gallons of white and maroon paint Frank will need, we need to calculate the total square footage for each color. Here's a step-by-step explanation:

1. Determine the square footage of the floor and ceiling that will be painted white. Since they are the same size, we can calculate the area of one and multiply it by 2.
2. Determine the square footage of all the walls that will be painted maroon. Calculate the area of each wall and sum them up.
3. Determine the square footage of the door that will be painted white. Subtract this value from the total maroon wall area.
4. Divide the total square footage of the white and maroon surfaces by 400 sq ft (coverage of one gallon) to find out how many gallons are needed for each color.

After calculating the areas and the number of gallons needed, compare the results with the given options (A, B, C, or D). Keep in mind that we don't have the specific dimensions for Frank's room, but following these steps will help you solve the problem once you have the necessary measurements.

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The value of angle x = 114°.

From the figure, it is clear that The interior angle of a triangle is 39°, by the law of opposite angle.

The sum of the interior angle of a triangle is 180°

37° + 39° + ∠unknown1 = 180°

∠unkonown1 = 180° - 37° - 39°

∠unknown1 = 104°

The sum of the exterior angle and the interior angle is 180°.

∠unknown2+ ∠unknown 1= 180°

∠unknown2 = 180° - 104°

∠unknown2 = 76°

The sum of the interior angle of a triangle is 180°

∠unknown3 + ∠unknown2 + 38 = 180

∠unknown3 + 76° + 38 = 180

∠unknown3= 66°

The sum of the exterior angle and the interior angle is 180°.

∠X + <unknown3 = 180°

∠X = 180° - 66°

∠X = 114°

The value of the angle x is 114°.

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To find out how long Sara studied in all, we need to add the time she studied before dinner and after dinner.

Sara studied for 3 1/2 hours before dinner and 45 minutes after dinner.

We can convert 3 1/2 hours to minutes by multiplying it by 60:

3 1/2 hours = 3 × 60 + 30 = 180 + 30 = 210 minutes

So, Sara studied for 210 minutes before dinner and 45 minutes after dinner.

To find the total time, we add these two values:

Total time = 210 minutes + 45 minutes = 255 minutes

Therefore, Sara studied for 255 minutes in all.

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Based on the results of the survey, a circle graph would have 6 sectors that are labeled as 10% or less and 3 sectors that are labeled as more than 10%.

The survey results are as follows:

there are a total of 10 categories, 6 of which (Food, Drinks, Clothing, Shoes, Personal Care, and Other) have percentages equal to or below 10% while the other 4 (Accessories, Video Games, Electronics, and Other) have percentages above 10%.

Since the complete circle in a circle graph reflects 100% of the data, categories with a percentage of less than or equal to 10% will have smaller sectors in the graph than those with a percentage of more than 10%.

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The transformations that have occurred from the original parent function ()=√య to the given function ()=2√x+4 are: vertical stretch by a factor of 2 and a vertical shift upward by 4 units.

The transformations that have occurred from the original parent function ()=√x to the given function ()=2√x+4 are: vertical stretch by a factor of 2 and a vertical shift upward by 4 units. The square root function (√x) has been multiplied by 2, resulting in a steeper curve, and then shifted vertically upwards by 4 units.

From the original cupcake company function g(x), the new function h(x)=2√x+2య+4 involves additional transformations. It starts with the transformations from part (a), which are a vertical stretch by a factor of 2 and a vertical shift upward by 4 units.

Annndditionally, the function is further transformed by a horizontal compression by a factor of 1/2, achieved by dividing the x-values by 2. Finally, a vertical shift upward by 2 units is applied. These transformations modify the shape, position, and scale of the original function to represent the changes in Mrs. Galicia's cupcake company's earnings.

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Answer:

  area ≈ 9.219 square units

Step-by-step explanation:

You want the approximate area under the curve y = -1/2x² +x +5 on the interval [1.5, 4] using 5 trapezoids.

The interval can be divided into 5 intervals of width ...

  (4 -1.5)/5 = 2.5/5 = 0.5

The "bases" of each trapezoid will be the function values at the ends of the intervals, for example, at x=1.5 and x=2. The "height" of each trapezoid is the width of the sub-interval, 0.5.

The area formula for a trapezoid applies:

  A = 1/2(b1 +b2)h

  A = 1/2(f(x) +f(x +0.5))·0.5 . . . . . for x = 1.5, 2, 2.5, 3, 3.5

The sum of the areas is computed in the attachment as ...

  area under the curve = 9.21875

__

Additional comment

The value of the integral is 445/48 ≈ 9.2708333...

The probability of being assigned to a desk in the front row exactly four times over the course of seven days is approximately 0.008, or 0.8%.

There are a total of 16 desks in the classroom, arranged in 4 rows and 4 columns. The probability of being assigned to a desk in the front row is 4/16 = 1/4, since there are 4 desks in the front row.

To calculate the probability of being assigned to a front-row desk exactly 4 times over the course of 7 days, we can use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)

where X is the random variable representing the number of times you are assigned to a front-row desk, n is the number of trials (in this case, 7), k is the number of successes (being assigned to a front-row desk), p is the probability of success on each trial (1/4), and (n choose k) represents the number of ways to choose k successes out of n trials, which is given by the binomial coefficient formula:

(n choose k) = n! / (k! * (n-k)!)

where ! represents the factorial function.

Using this formula, we get:

P(X = 4) = (7 choose 4) * (1/4)^4 * (3/4)^3

P(X = 4) = (35) * (1/256) * (27/64)

P(X = 4) ≈ 0.008

Therefore, the probability of being assigned to a desk in the front row exactly four times over the course of seven days is approximately 0.008, or 0.8%.

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Base on the word problem, Bernard is currently 21.25 years old. When Hector turns 18, he will be (18 - 17) = 1 year older than his current age. At that time, Bernard will be (21.25 + 1) = 22.25 years old.

Let's start by assigning variables to represent the current ages of Bernard and Hector. Let B be Bernard's current age and H be Hector's current age. Then we can write two equations based on the given information:

"When Bernard was as old as Hector is now, Bernard's age was 4 times Hector's age then." This means that Bernard is currently (B - H) years older than Hector, and that the age difference between them has remained constant over time. So, we can write: B - (B - H) = 4(H - (B - H)).

Simplifying this equation, we get: B - B + H = 4(2H - B)

Simplifying further, we get: 5H - 4B = 0, or B = (5/4)H.

"When Hector will be as old as Bernard is now, the sum of their ages will be 51." This means that when Hector is (B - H) years older than his current age, their sum of ages will be 51. So, we can write: B + (B - H + (B - H)) = 51.

Simplifying this equation, we get: 3B - 2H = 51.

Now we have two equations with two variables. We can substitute the expression for B from the first equation into the second equation, and solve for H:

3B - 2H = 51

3(5/4)H - 2H = 51

(15/4)H = 51

H = 17

So, Hector is currently 17 years old. To find out how old Bernard will be when Hector turns 18, we can use the expression we found earlier for B in terms of H:

B = (5/4)H

B = (5/4)(17)

B = 21.25

So, Bernard is currently 21.25 years old. When Hector turns 18, he will be (18 - 17) = 1 year older than his current age. At that time, Bernard will be (21.25 + 1) = 22.25 years old.

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I selected Josue as the dog-walker.

Frequency tables:

Morning dogs:

Weight (lbs) Frequency

10-19 2

20-29 5

30-39 2

40-49 1

Afternoon dogs:

Weight (lbs) Frequency

7-16 4

17-26 2

27-36 1

37-46 1

47-56 2

The median of the morning (AM) group is 26.5 lbs. The median of the afternoon (PM) group is 23 lbs.

The first quartile (Q1) of the morning (AM) group is 16.25 lbs. The first quartile (Q1) of the afternoon (PM) group is 9.5 lbs.

The third quartile (Q3) of the morning (AM) group is 34.75 lbs. The third quartile (Q3) of the afternoon (PM) group is 38.5 lbs.

Comparative box plot:

yaml

Copy code

Morning dogs:               Afternoon dogs:

 13     |                      7     |

        |                              |

 16     |                      9     |

        |                              |

 21     |                     11     |

        |                              |

 25     |                     15     |

        |                              |

 26     |                     27     |

        |                              |

 28     |                     34     |

        |                              |

 30     |                     35     |

        |                              |

 35     |                     39     |

        |                              |

 38     |                     43     |

        |                              |

        |                     44     |

        +------------------------------+

          1    2    3    4    5    6

              Group

Morning dogs:

Min: 13

Q1: 16.25

Median: 26.5

Q3: 34.75

Max: 38

Afternoon dogs:

Min: 7

Q1: 9.5

Median: 23

Q3: 38.5

Max: 44

The interquartile range (IQR) of the morning (AM) group is 18.5 lbs. The IQR of the afternoon (PM) group is 29 lbs.

Based on the comparative box plot and the IQRs, the morning group of dogs would be easier to walk as one group. This is because the morning group has a smaller IQR, indicating that the weights of the dogs are more similar to each other. The afternoon group has a larger IQR, indicating that the weights of the dogs are more spread out, which could make it more difficult to walk them as a group.

1. JosueSara

2. Frequency table

3. Median of the Morning Group: 26.5, Median of the Afternoon Group: 18.5

4. Q1 of the Morning Group: 17.5, Q1 of the Afternoon Group: 10.5

5. Q3 of the Morning Group: 32.5, Q3 of the Afternoon Group: 36.5

6. Comparative Boxplot blue is morning dogs and red is afternoon dogs.

7. IQR of the Morning Group: 15, IQR of the Afternoon Group: 26

8. Based on the comparative box plot and the IQRs, the morning group of dogs would be easier to walk as one group.

What is boxplot?

A box plot, also known as a box-and-whisker plot, is a graphical representation of the distribution of a dataset. It displays summary statistics and provides a visual summary of the data's key characteristics.

1. Which dog-walker did you select?

JosueSara

I selected Sara.

2. Create frequency tables to represent the morning and afternoon dogs as two sets of data. Group the weights into classes that range 10 pounds.

Morning Dogs Frequency Table:

Weight Range Frequency

10-19                        2

20-29                        4

30-39                        4

Afternoon Dogs Frequency Table:

Weight Range Frequency

0-9                               1

10-19                       3

20-29                       2

30-39                       2

40-49                       1

50-59                       1

3. What is the median of the morning (AM) group? What is the median of the afternoon (PM) group?

Median of the Morning Group: 26.5

Median of the Afternoon Group: 18.5

4. What is the first quartile (Q1) of the morning (AM) group? What is the first quartile (Q1) of the afternoon (PM) group?

Q1 of the Morning Group: 17.5

Q1 of the Afternoon Group: 10.5

5. What is the third quartile (Q3) of the morning (AM) group? What is the third quartile (Q3) of the afternoon (PM) group?

Q3 of the Morning Group: 32.5

Q3 of the Afternoon Group: 36.5

6. Create a comparative box plot for the morning and afternoon dogs, and label each with its five-number summary.

Morning Dogs:

Min: 13

Q1: 17.5

Med: 26.5

Q3: 32.5

Max: 38

Afternoon Dogs:

Min: 7

Q1: 10.5

Med: 18.5

Q3: 36.5

Max: 55

7. What is the interquartile range (IQR) of the morning (AM) group? What is the interquartile range (IQR) of the afternoon (PM) group?

IQR of the Morning Group: 15

IQR of the Afternoon Group: 26

8. The average weights of the dogs are the same for the morning and afternoon groups. But based on your comparative box plot and the IQRs of the two groups, which group of dogs do you think would be easier to walk as one group? Why?

Based on the comparative box plot and the IQRs, the morning group of dogs would be easier to walk as one group. This is because the morning group has a smaller interquartile range (IQR) of 15 compared to the afternoon group's IQR of 26. A smaller IQR indicates that the weights of the dogs in the morning group are more clustered together.

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The function that describes how many lanterns Tisha can buy is F(x) = 18x + 64, with a domain of {0, 1, 2, 3, 4} and a range of {64, 82, 100, 118, 136}.

Function to describe how many lanterns Tisha can buy: F(x) = 18x + 64.

Domain: {0, 1, 2, 3, 4, 5} (since Tisha can buy up to 4 additional packages of lanterns, plus the original 64 lanterns).

Range: {64, 82, 100, 118, 136, 154} (each additional package of lanterns has 18 lanterns, so the total number of lanterns Tisha can buy is a multiple of 18 added to 64).

Option (a) F(x) - 18x + 64 has the correct formula but an incorrect domain. Tisha can buy 0 packages of lanterns, so the domain should include 0.

b) For option b, the function is F(x) = 18x + 64, where x represents the number of packages of lanterns Tisha buys. The reasonable domain for this function is {0, 1, 2, 3, 4, 5}, since Tisha can buy up to 4 packages and may choose not to buy any, resulting in x = 0. The range for this function is {64, 82, 100, 118, 136, 154}, which represents the total number of lanterns Tisha can have after buying x packages of 18 lanterns each, starting from the initial 64 lanterns she already has.

Option (c) F(x) - 64x + 18 has an incorrect formula. Tisha starts with 64 lanterns, so the constant term should be 64, not 18.

Option (d) F(x) = 64x + 18 has the correct formula, but the domain is incorrect. Tisha can only buy up to 4 packages, so the domain should be {0, 1, 2, 3, 4}, not just 5.

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The concentration of salt in the tank at the end of 10 minutes is 0.01 pounds per gallon

We can use the formula:

(concentration of salt in tank) * (gallons of water in tank) = (total pounds of salt in tank)

To solve this problem. At the beginning, the tank contains 500 gallons of salt-free water, so the total pounds of salt in the tank is 0. After 10 minutes, 20 gallons of brine have entered the tank, and 20 gallons of the mixture have left the tank. As a result, the amount of water in the tank remains constant at 500 gallons.

The amount of salt that enters the tank in 10 minutes is:

(0.25 lb/gal) * (2 gal/min) * (10 min) = 5 lb

The total pounds of salt in the tank after 10 minutes is:

0 + 5 = 5 lb

Therefore, the concentration of salt in the tank at the end of 10 minutes is

(concentration of salt in tank) * (500 gallons) = 5 lb

Solving for the concentration of salt in the tank, we get:

concentration of salt in tank = 5 lb / 500 gallons

Simplifying this expression, we get:

concentration of salt in tank = 0.01 lb/gal

Therefore, the concentration of salt in the tank at the end of 10 minutes is 0.01 pounds per gallon.

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A. angle CXD = 140 degrees

Angle XCD and Angle XDC are congruent since the triangle is isosceles. Remember that the sum of the interior angles of a triangle is 180 degrees.

20 + 20 + x = 180

x = 140

B. 18 sides

To find the number of sides of the polygon, we need to know the measure of one interior angle. One interior angle is angle BCD. We can easily find the measure of this angle because it is on a straight angle of which we are given part of (angle XCD).

Angle XCD + Angle BCD = 180

20 + BCD = 180

BCD = 160

Now that we know the measure of an interior angle, we can use the formula to find the measure of an interior angle and algebraically solve for the number of sides.

[ (n - 2) x 180 ] / n = 160

(n - 2) x 180 = 160n

180n - 360 = 160n

-360 = -20n

n = 18 sides

C. 2880 degrees

The formula for the sum of the interior angles of a regular polygon is (n - 2) x 180, where n is the number of sides.

(18 - 2) x 180

16 x 180

2880

D. 140 degrees

If angle XCD is 20 degrees, then angle BED is also 20 degrees. Angle BED and Angle BEF make up one of the interior angles of the regular polygon. We know that one interior angle is equal to 160 degrees.

Angle BED + Angle BEF = 160

20 + BEF = 160

BEF = 140

Hope this helps!

So, the final answer is:

(1/2)(-√(4 - x^2) / x) + C. To evaluate the integral ∫dx / (x^2√(4-x^2)), we will use the trigonometric substitution x = 2sin(θ). This substitution is chosen because it simplifies the expression under the square root, as 4 - x^2 becomes 4 - 4sin^2(θ) which can be factored into 4cos^2(θ).

Now, we need to find dx in terms of dθ. Differentiating x with respect to θ, we get:

dx/dθ = 2cos(θ) => dx = 2cos(θ)dθ

Substituting x = 2sin(θ) and dx = 2cos(θ)dθ into the integral:

∫(2cos(θ)dθ) / ((2sin(θ))^2√(4(1-sin^2(θ))))
= ∫(2cos(θ)dθ) / (4sin^2(θ)√(4cos^2(θ)))

Simplifying the integral, we get:

= (1/2) ∫(cos(θ)dθ) / (sin^2(θ)cos(θ))
= (1/2) ∫dθ / sin^2(θ)

Now, use the identity csc^2(θ) = 1/sin^2(θ) and integrate:

= (1/2) ∫csc^2(θ) dθ
= (1/2)(-cot(θ)) + C

To find cot(θ), we draw a right triangle with the opposite side x, the adjacent side √(4 - x^2), and the hypotenuse 2:

cot(θ) = adjacent / opposite = √(4 - x^2) / x

So, the final answer is:

(1/2)(-√(4 - x^2) / x) + C

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Answer:

Step-by-step explanation: