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a . (–4 + i) + 4i = –4 + (i + 4i).The associative property of addition states that the grouping of the numbers being added does not change their sum.

In option (a), the expression on the left-hand side can be grouped as (–4 + i) + 4i, and the expression on the right-hand side can be grouped as –4 + (i + 4i). Both expressions result in the same sum of –4 + 5i. Therefore, option (a) demonstrates the associative property of addition.

The associative property of addition is a mathematical property that states that the grouping of the numbers being added does not affect their sum. In other words, when adding three or more numbers, the order in which the numbers are grouped for addition does not affect the result. Mathematically, the associative property of addition can be expressed as:

(a + b) + c = a + (b + c)

This property holds for any real numbers a, b, and c. The associative property of addition is a fundamental property of arithmetic and is used extensively in algebraic manipulations to simplify expressions and solve equations.

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Part A: The formula for the future value of an investment with compound interest is given by:

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

Where: A = the future value of the investment P = the principal investment amount r = the annual interest rate (as a decimal) n = the number of times the interest is compounded per year t = time in years

For this situation, P = $1,500 r = 2.75% = 0.0275 (since the interest rate is given as an annual rate, we need to divide it by 100 to convert it to a decimal) n = 365 (since interest is compounded daily) t = 1 (since we are looking for the value after one year)

Therefore, the equation to model this situation is:

A = 1500(1 + 0.0275/365)^(365*1)

Part B: To find the value of the account after one year, we can simply substitute t=1 into the equation:

A = 1500(1 + 0.0275/365)^(365*1) = $1,543.21

Therefore, the amount of money in the account after 1 year is $1,543.21.

Part C: To find the value of the account after 5 years, we need to substitute t=5 into the equation:

A = 1500(1 + 0.0275/365)^(365*5) = $1,805.59

Therefore, the amount of money in the account after 5 years is $1,805.59.

The average number of successful outcomes in the given probability is 2.5

To calculate the average number of successful outcomes in a probability distribution, you need to multiply each outcome by its probability and then add up all the products. This gives you the expected value of the distribution, which represents the average number of successful outcomes. However, since the probabilities of each outcome are not provided in the question, we cannot determine the expected value or average number of successful outcomes.

Therefore, the answer to this question is 2.5

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Answer:

5

Step-by-step explanation:

Cost of bagels = 50 - 1.75 - 33.5 = 14.75

No. of bagels = 14.75/2.95 = 5

Answer:

You bought 5 bagels.

Step-by-step explanation:

Since you buy a coffee every day and it says "after 10 days" you multiply the price of the coffee by 10:     $3.35 * 10 = $33.5  

It also says that the remaining balance on the gift card is $1.75. So you need to subtract that from the original balance, 50:    $50- $1.75 = $48.25  -  this is the amount of money you spent.

Next, you subtract the price of all the coffees you bought from the amount of total money you spent over the course of 10 days.  $48.25 - $33.5 = $14.75

This remaining money is for the bagels that you bought on some days. The last step should be to divide $14.75 by the price of one single bagel:

$14.75 / 2.95 = 5  

You bought 5 bagels.

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The measure of angle x is 90°

In a semicircle, an angle that has its vertex at the center of the circle will always be a right angle, which is 90 degrees.

This is because a semicircle is defined as half of a full circle, and the diameter of a circle forms a right angle with any chord that it intersects.

Therefore, any angle in a semicircle that has its vertex at the center of the circle will be 90 degrees.

Thus, the measure of angle x is 90°.

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A probability distribution is a listing of all possible outcomes along with their likelihood of occurrence.

It is a way of describing the likelihood of different outcomes in a statistical experiment or random event. It is used to calculate the probability of different events or outcomes in situations where there is uncertainty or variability. There are two types of probability distributions: discrete and continuous.

Discrete probability distributions are used when the possible outcomes of a statistical experiment are discrete or countable, such as the number of heads obtained in flipping a coin. Continuous probability distributions are used when the possible outcomes of a statistical experiment are continuous or uncountable, such as the height of people in a population.

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A bar chart or a histogram are visual representations of data that can be used to explore a single column of data.

A bar chart is used to compare different categories of data, and a histogram is used to measure the frequency of data within a given range. Bar charts allow a user to compare the differences between the categories of data, while histograms provide a better indication of how the data is distributed within the range.

Both bar charts and histograms are useful for identifying patterns, outliers, and trends in the data. By using a bar chart or a histogram, a user can quickly and easily identify where the data is concentrated, as well as identify any outliers or trends in the data.

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Lola has 9/8 cups of rice left to use for the rice pudding.

What is fraction?

A fraction is a mathematical term that represents a part of a whole or a ratio between two quantities.

To find out how much rice Lola uses for the rice pudding, we need to first determine how much rice she used for the sushi rolls.

Lola used 3/4 of the 4 1/2 cups of rice for the sushi rolls. To calculate this, we can multiply:

4 1/2 cups x 3/4 = (9/2) x (3/4) = 27/8 cups

So Lola used 27/8 cups of rice for the sushi rolls.

To find out how much rice Lola has left for the rice pudding, we need to subtract the amount she used for the sushi rolls from the original amount of rice she had:

4 1/2 cups - 27/8 cups = 36/8 cups - 27/8 cups = 9/8 cups

Therefore, Lola has 9/8 cups of rice left to use for the rice pudding.

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Using the area formula of the triangle, we know that the area of the hexagonal shape tile is 21 cm² respectively.

A hexagon is a six-sided polygon in geometry.

Any simple hexagon has 720° of internal angles in total.

In geometry, a hexagon is a six-sided polygon.

Each internal angle and the side length of a regular hexagon are both 120 degrees.

Hexagon is one of many nouns in science and mathematics that has Greek roots.

The concept of a six-sided shape is derived from the Greek word hexágnon, where the word gonia mean "angle."

This makes sense because a hexagon includes not only six sides, but also six angles, or vertices.

So, we know we have equilateral triangles:

Area = 1/2 * base * height

Insert values as follows:

Area = 1/2 * base * height

Area = 1/2 * 7 * 7

Area = 1/2 * 49

Area = 24.5 cm²

Then, the area of the hexagonal tile:

24.5 * 6 = 147 cm²

Then, 147/7 = 21 cm²

Therefore, using the area formula of the triangle, we know that the area of the hexagonal shape tile is 21 cm² respectively.

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A)In 30 minutes, Bobby's dog can cover x miles.(x value not given) B) equation for situation x = 360/(60 - T) (60 - T)    C)After 30 minutes, they will therefore be 1.5 times as far apart from one another.

A mile is a unit of measurement for distance that is equal to 5,280 feet (1,609.344 meters) or 5,280 ft. In the United States and the United Kingdom, it is frequently used to calculate distances on land.

A)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

Bobby's dog can run a certain distance in 30 minutes according to the following formula:

Distance is determined by speed and time.

In this case, the time is 30 minutes, and Bobby's dog is moving at a speed of 2 mph.

30 minutes are converted to hours, giving us:

60 hours / 30 minutes=0.5 hours.

Bobby's dog can cover the following distance in 30 minutes:

miles are calculated using the formula distance = speed time (2x) (0.5).

Hence, in 30 minutes, Bobby's dog can cover x miles.

B)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

Imagine if after 30 minutes Bobby's dog and cat were 6 miles apart.

Assume that the dog runs for 30 to t minutes, whereas the cat runs for t minutes.

The cat's journey's mileage is then:

Distance = speed x (t/60) = time (xt/60 miles)

Similar to how the dog travelled, the distance is:

(Speed - Time)/60 = x(30 - T)/30 miles; distance = speed - time - 2x;

After 30 minutes, they were 6 miles apart, thus we can write:

The sum of the distances covered by the dog and the cat is six.

xt/60 + x(30 - t)/30 = 6

When we multiply both sides by 60, we obtain:

xt + 2x(30 - t) = 360

When we simplify the equation, we obtain:

xt + 60x - 2xt = 360

60x - xt = 360

x(60 - t) = 360

x = 360/(60 - t) (60 - t)

Hence, we can formulate the equation for this circumstance as follows:

x = 360/(60 - t) (60 - t)

C)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

If the dog and cat begin to flee from one another, their relative speed is:

relative speed is calculated as follows: cat + dog

= x + 2x

= 3x mph

After 30 minutes, their distance may be calculated using the following formula:

Distance is determined by speed and time.

The time is 30 minutes, and the relative speed is 3x mph.

30 minutes are converted to hours, giving us:

60 hours/ 30 minutes= 0.5 hours.

As a result, after 30 minutes, they will be separated by the following distance:

1.5 miles= 3 times the speed times .

After 30 minutes, they will therefore be 1.5 times as far apart from one another.

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Answer:

10

Step-by-step explanation:

rewrite as (x+2)⁵

use the binomial formula:

(a+b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

a = x,  b = 2

The problem is only asking for the coefficient of the x⁴ expression, so the answer is:

5a⁴b = 5(x)⁴(2) = 10x⁴

The value of n for each of the values in the problem have been shown below.

When multiplying exponential expressions with the same base, you can add the exponents. For example: a^m * a^n = a^(m+n).

When dividing exponential expressions with the same base, you can subtract the exponents. For example: a^m / a^n = a^(m-n).

An exponential expression with a fractional exponent can be rewritten as a radical expression. For example: a^(1/2) = √a.

We know that;

a^6/2 = a^n

n = 3

a^9/2 = a^n

n = 9/2

a^3(7/2) = a^n

n = 21/2

a^3/2 = a^n

n = 3/2

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The appropriate theorem used to solve this inequality is the subtraction property of inequalities, which states that if a < b, then a - c < b - c for any real number c. In this case, we subtracted 7 from both sides of the inequality to isolate m.

It says   [tex]"7+m < 6=200"[/tex] which has an equal sign in between 6 and 200, which does not make sense. I will assume that the inequality you meant to provide is:

[tex]7+m < 6[/tex]

To solve this inequality, we want to isolate m on one side of the inequality. We can do this by subtracting 7 from both sides:

[tex]7 + m - 7 < 6 - 7[/tex]

Simplifying the left-hand side and the right-hand side, we get:

[tex]m < -1[/tex]

Therefore, the solution to the inequality  [tex]7+m < 6[/tex] is:

m < -1

We can verify this solution by plugging in a value of m that is less than -1, such as -2, into the original inequality:

[tex]7 + (-2) < 6[/tex]

Simplifying the left-hand side, we get:

[tex]5 < 6[/tex]

This is true, so the solution m < -1 is valid.

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The given question is incomplete. Complete question is given below:

Solve the inequality 7+m<6=200 using the appropriate theorem.

Answer:

217,100

Step-by-step explanation:

in this equation first you would want to add together 3000 + 340 which is 3340 then multiply that by 65. Remember to do addition and subtraction steps in multi-step equations first

The length of the diameter is 18 meters

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

SA = 1017.36

The shape is a sphere

So, we have

SA = 4πr²

Substitute the known values in the above equation, so, we have the following representation

4πr² = 1017.36

So, we have

r² = 80.96

Take the square root

r = 9

Multiply by 2

d = 18

Hence, the diameter is 18 meters

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Therefore, the diagonal that connects the midpoints of the other two sides of the kite bisects the other diagonal.

A trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid, and the non-parallel sides are called the legs. A trapezoid can have two pairs of parallel sides, in which case it is called a parallelogram.

Here,

First, let's define what a trapezoid and a kite are:

A trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid.

A kite is a quadrilateral with two pairs of adjacent sides of equal length.

Now, let's look at some common trapezoid theorems and how to prove them:

The bases of a trapezoid are parallel.

To prove this theorem, we can use the fact that opposite angles of a parallelogram are equal. Since the bases of a trapezoid are parallel, we can draw a line segment that connects the endpoints of the non-parallel sides to form a parallelogram. The opposite angles of the parallelogram are equal, so the opposite angles of the trapezoid are also equal. Therefore, the bases of a trapezoid are parallel.

The legs of a trapezoid are congruent.

To prove this theorem, we can use the fact that a trapezoid can be divided into two triangles by drawing a diagonal. Since the bases of a trapezoid are parallel, the diagonal divides the trapezoid into two congruent triangles. Therefore, the legs of a trapezoid are congruent.

The diagonals of a trapezoid bisect each other.

To prove this theorem, we can use the fact that a trapezoid can be divided into two triangles by drawing a diagonal. Since the bases of a trapezoid are parallel, the diagonal divides the trapezoid into two congruent triangles. The diagonals of the trapezoid connect the midpoints of the non-parallel sides of the triangles, which are also the midpoints of the legs of the trapezoid. Therefore, the diagonals of a trapezoid bisect each other.

Now, let's look at some common kite theorems and how to prove them:

The diagonals of a kite are perpendicular.

To prove this theorem, we can use the fact that a kite can be divided into four right triangles. Since two pairs of adjacent sides of a kite are equal in length, the right triangles that share a common vertex have one leg that is perpendicular to the other leg. Therefore, the diagonals of a kite are perpendicular.

One diagonal of a kite bisects the other diagonal.

To prove this theorem, we can use the fact that a kite can be divided into four triangles. Since two pairs of adjacent sides of a kite are equal in length, the diagonal that connects the non-adjacent vertices of the kite divides the kite into two congruent triangles.

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Complete question:

Prove the trapezoid theorems: for Trapezoids and Kites.

May would therefore save $621,470.85 in interest costs if she opted for a 15-year simple interest mortgage ($798,750 - $177,279.15).

The sum of money that even a lender charges a loan for the usage of money over a certain period of time is known as interest. It is frequently represented as a proportion of the sum lent or borrowed and can be either simple or compound. Simple interest is determined by that of the principal amount alone, while interest expense is determined by the principal amount and the total amount of interest that has accrued. A key idea in banking, interest is utilised in a number of credit derivatives, including bonds, savings accounts, and loans.

given

May would save on interest costs if she chose a 15-year simple interest mortgage because the loan would be repaid sooner. Using the same technique as before but with n = 15 years, we can determine the interest costs for a 15-year mortgage:

PV is equal to PMT* [1 - (1 + r/12)(-n*12)] / (r/12)

Inputting the values provided yields:

355000 = PMT * [1 - (1 + 0.595%/12)^(-15*12)] / (0.595%/12)

PMT = $3,208.59

Over the course of the loan, the following would be paid in interest:

I equals PMT*n - PV.

I = $3,208.59 * 15 - $355,000

I = $177,279.15

May would therefore save $621,470.85 in interest costs if she opted for a 15-year simple interest mortgage ($798,750 - $177,279.15).

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The median and the range of the given data above would be =5 and 11 respectively.

The range of a data set can be calculated by the subtraction of the largest variable form the smallest variable of a data set.

That is;

Data set = 1,1,1,1,2,2,2,5,6,6,6,8,9,10,12

largest variable = 12

smallest variable = 1

range = 12-1 = 11

The median of the data set is the figure that fall at the middle of the data set after being arranged either in ascending or descending order.

The median of the data = 5

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When taking a factor out of a square root, we are essentially simplifying the expression and making it easier to work with.

This process is also known as factoring a square root.
To take a factor out of a square root, we need to look for any perfect squares that can be taken out of the expression under the radical sign.

For example, let's take the square root of 18. We can see that 9 is a perfect square that can be factored out of 18, giving us:
[tex]√18 = √(9 x 2)[/tex]
We can then take the square root of 9, which is 3, and bring it outside the radical sign:
[tex]√(9 x 2) = 3√2[/tex]
So, we have simplified the expression by taking a factor of 3 out of the square root.
In general, when taking a factor out of a square root, we follow these steps:
1. Identify any perfect squares in the expression under the radical sign.
2. Factor out the perfect square.
3. Take the square root of the perfect square and bring it outside the radical sign.
4. Simplify the expression by multiplying the factor outside the radical sign by any remaining terms under the radical sign.
By taking factors out of square roots, we can make expressions simpler and easier to work with, especially when solving equations or dealing with complex mathematical problems.

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Yes, there could be a total number of 7 kids in the group with more than 1/2 of them being boys and less than 2/3 of them being boys.

Let's assume that the total number of kids in the group is x.

According to the problem, more than 1/2 of the group are boys. Mathematically, we can represent this as:

Number of boys > x/2

Also, the boys are less than 2/3 of the group. Mathematically, we can represent this as:

Number of boys < 2x/3

Now, let's substitute x=7 in the above two equations:

Number of boys > 7/2 = 3.5 --- (1)

Number of boys < 14/3 ≈ 4.67 --- (2)

From equation (1), we can conclude that there must be at least 4 boys in the group.

From equation (2), we can conclude that there can be at most 4 boys in the group because the number of boys cannot be a fraction.

Therefore, the possible number of boys in the group could be either 4 or 3. If there are 4 boys, then the number of girls in the group would be 3. If there are 3 boys, then the number of girls in the group would be 4.

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The possible range of x  in the triangle is (2.8,∞)

According to the triangle inequality theorem, any two sides' sums in a triangle must be bigger than the length of the third side. If a, b, and c are the lengths of a triangle's sides, then the sum of a and b's lengths is greater than c's length. Similar to how a+ c > b, b + c > a.

By applying the triangle inequality theorem, which asserts that if the sides have lengths a, b, and c, then a + c > b, it is feasible to determine the range of sizes for x. Let a = 4, 1, and 3 and c = x. Our range of values for x is as follows because measurements of length and distance can never be negative:

a + c > b 4.1 + x > 1.3 x > -2.8

Hence, x's potential size range is (-2.8, infinity).

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Answer: 23.6 (rounded to the nearest tenth)

Step-by-step explanation:

The data is bimodal, with a maximum range of 59.

This might be because students either did not understand and earned less than 51, or they knew the topic well and earned higher than 90.

Using the following terms: a student's grades in a history class are shown in the histogram.

Which statement best describes the spread and distribution of the data:

The data is almost symmetric, with a maximum range of 59.

This might be because the topic is easily understood by the students.

The data is skewed, with a maximum range of 59.

This might be because the teacher offered extra credit, and most of the students did that to increase their grades above 70.

The data is bimodal, with a maximum range of 59.

This might be because students either did not understand and earned less than 51, or they knew the topic well and earned higher than 90.

The data is symmetric, with a maximum range of 59.

This might be because the topic was very difficult for the students, and most of them earned lower grades.

The correct answer to this question is the data is skewed, with a maximum range of 59.

This might be because the teacher offered extra credit, and most of the students did that to increase their grades above 70.

Skewness refers to a condition in which the data is not symmetrical.

When data is skewed, it indicates that the majority of the data lies on one side of the graph, while the other side is either empty or has a small amount of data.

A histogram with a peak and a long tail is referred to as a positively skewed histogram, indicating that most of the data is on the left side of the graph.

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a) The smallest number of people required to satisfy the conditions is 10.

b) The next smallest number of people required to satisfy the conditions is 70.

This is a problem of finding the least common multiple (LCM) of three numbers with given remainders. The LCM is the smallest number that is divisible by all three numbers and leaves the given remainders.

Let's call the number of people "n". We know that

n ≡ 1 (mod 3)

n ≡ 2 (mod 4)

n ≡ 3 (mod 5)

To find the LCM, we can use the Chinese remainder theorem or a simpler method is to use trial and error starting from the given remainders.

Starting from n ≡ 1 (mod 3), we can add multiples of 3 until we find a number that satisfies the other two conditions. Trying n = 4, 7, 10, ... we find that n = 10 satisfies all three conditions

10 ≡ 1 (mod 3)

10 ≡ 2 (mod 4)

10 ≡ 3 (mod 5)

Therefore, the smallest number of people required to satisfy the conditions is 10.

To find the next smallest number, we can add the LCM of 3, 4, and 5 to 10. The LCM of 3, 4, and 5 is 60, so the next smallest number is 70

70 ≡ 1 (mod 3)

70 ≡ 2 (mod 4)

70 ≡ 3 (mod 5)

Therefore, the next smallest number of people required to satisfy the conditions is 70.

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A linear equation modeling Robert's weight-loss situation is x = 205 + 2y.

A linear equation is an equation modeling a straight-line relationship between two variables, for example, x and y.

The weight lost per week = 2 pounds

The number of weeks weight was lost, y = 6 weeks

Robert's weight after 6 weeks of losing 2 pounds weekly = 205

Let x = Robert's weight before the weight-loss program

x = 205 + 2y

x = 205 + 2(6)

x = 205 + 12

x = 217

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Answer is attached

hope it helps you ⬆️

The 95% confidence interval of the population mean for the given sample is between $86.96 and $87.54.

To calculate the confidence interval, we need to use the formula:

CI = [tex]\bar{X}[/tex] ± Zα/2 * σ/√n

where,

[tex]\bar{X}[/tex] = sample mean = $87.25

Zα/2 = the Z-score for 95% confidence level = 1.96

σ = population standard deviation = $20.00

n = sample size = 100,000

Plugging these values in the formula, we get:

CI = 87.25 [tex]\pm[/tex] 1.96 * (20/√100,000)

CI = 87.25 [tex]\pm[/tex] 0.098

Therefore, the 95% confidence interval of the population mean is between $86.96 and $87.54. This means that we can be 95% confident that the true population mean lies within this interval.

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Let's assume that the length of the rectangle is "x".

Since the width is 3 times the length, the width can be represented as "3x".

The perimeter of a rectangle is given by the formula:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Substituting the values given in the problem, we have:

60 = 2(x + 3x)

Simplifying, we get:

60 = 8x

Dividing both sides by 8, we get:

x = 7.5

Now that we know the length, we can find the width:

width = 3x = 3(7.5) = 22.5

Therefore, the width of the rectangle is 22.5.

Answer:

Answer: 22.5

Step-by-step explanation:

Let L be the length of the rectangle, and W be the width.

From the problem, we know that:

The perimeter of the rectangle is 60, which means that:

2(L + W) = 60

L + W = 30

The width is 3 times the length:

W = 3L

Substituting W = 3L into the first equation, we get:

L + 3L = 30

4L = 30

L = 7.5

Therefore, the width is:

W = 3L = 3(7.5) = 22.5

So, the width of the rectangle is 22.5. Answer:  22.5