Explain in words how to write a rational equation

If your friend tells you that 2 times the first number is 13 more than the second number, the two numbers are 25 and 37.

Let's assume the first number is represented by x and the second number is represented by y. According to the given information, we can create two equations:

2x = y + 13 (Equation 1)

x + y = 62 (Equation 2)

We can use Equation 2 to solve for x:

x = 62 - y

We can substitute this value of x in Equation 1 to solve for y:

2(62 - y) = y + 13

124 - 2y = y + 13

124 = 3y + 13

111 = 3y

y = 37

Now that we have found y, we can substitute this value in Equation 2 to solve for x:

x + 37 = 62

x = 25

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The probability that someone is driving slower than 20 mph on the 405 freeway is approximately 7.08%.

We are given that the average speed on the 405 freeway is 42 mph and is normally distributed with a standard deviation of 15 mph.

To solve this problem, we need to use the standard normal distribution since we are given the mean and standard deviation of the speed on the 405 freeway.

We know that

[tex]Z=\frac{X-\mu}{\sigma}[/tex]

Here, X = 20

[tex]\mu=42[/tex]

[tex]\sigma=15[/tex]

Z=(20-42)/15

Z=-22/15

Z=-1.47

Using a standard normal distribution table

P(Z<-1.47)=0.0708

or 7.08%

Therefore, the probability that someone is driving slower than 20 mph on the 405 freeway is approximately 0.0708 or 7.08%.

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Yes, we can use z test.

The normal approximation to the binomial can be used for the hypothesis test of population proportion for option b but not for options a, c and d.

The Z-test is used to test the hypothesis about population parameters when the sample size is large and the population standard deviation is known or can be estimated from the sample.

For the case of a hypothesis test for a population proportion,

the Z-test can be used when the sample size is large enough to satisfy the conditions of normal approximation to the binomial distribution.

The conditions for normal approximation to the binomial distribution are:

The sample size, n, is large enough that np ≥ 10 and n(1-p) ≥ 10, where p is the population proportion.

The observations are independent.

Based on these conditions, we can determine whether or not the normal approximation to the binomial should be used for each of the following cases:

a. n=20 and H0:p=0.3.

In this case, np = 20 × 0.3 = 6 and n(1-p) = 20 × 0.7 = 14, both of which are less than 10. Therefore, the normal approximation to the binomial should not be used.

b. n=70 and H0:p=0.2.

In this case, np = 70 × 0.2 = 14 and n(1-p) = 70 × 0.8 = 56, both of which are greater than 10. Therefore, the normal approximation to the binomial can be used.

c. n=100 and H0 : p=0.08.

In this case, np = 100 × 0.08 = 8 and n(1-p) = 100 × 0.92 = 92, both of which are less than 10.

So, the normal approximation to the binomial should not be used.

d. n=150 and H0:p=0.01.

In this case, np = 150 × 0.01 = 1.5 and n(1-p) = 150 × 0.99 = 148.5, both of which are less than 10. Therefore, the normal approximation to the binomial can be used for the hypothesis test of population proportion for option b but not for options a, c and d.

In the cases where the normal approximation cannot be used, alternative methods like the exact binomial test or the chi-square test can be used.

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Assuming all conditions for inference are met, a 95 per cent confidence interval for the slope of the least-squares regression line is defined as the range of values within which we are 95 per cent confident that the true population slope lies.

It is calculated as the point estimate (the slope of the least-squares regression line) plus or minus the margin of error, which is determined by multiplying the standard error of the slope by the critical value from the t-distribution with n-2 degrees of freedom (where n is the sample size). This critical value is chosen such that 95 per cent of the t-distribution falls within the interval. Therefore, a larger sample size or a smaller standard error will result in a narrower confidence interval, while a smaller sample size or a larger standard error will result in a wider confidence interval.

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If the space is two-dimensional: c. 4

If the space is one-dimensional: b. 1

If the space is four-dimensional: a. 7

In rigid-body physics, the location of a rigid body can be specified by three coordinates (x, y, z) in three-dimensional space. The orientation of a rigid body can be specified by three angles (roll, pitch, yaw) or by a rotation matrix.

If the space is two-dimensional, the location of a rigid body can be specified by two coordinates (x, y). The orientation can be specified by one angle or by a 2x2 rotation matrix. Therefore, the total number of coordinates required is 3.

If the space is one-dimensional, the location of a rigid body can be specified by one coordinate (x). Since there is only one dimension, there is no need to specify orientation. Therefore, the total number of coordinates required is 1.

If the space is four-dimensional, the location of a rigid body can be specified by three coordinates (x, y, z) as in three-dimensional space. The orientation can be specified by four parameters, such as quaternions, which require four coordinates. Therefore, the total number of coordinates required is 7.

So, the answers are:

If the space is two-dimensional: c. 4

If the space is one-dimensional: b. 1

If the space is four-dimensional: a. 7

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Kayla ran a total of 3 and 1/3 miles last week.

Kayla ran 2/3 miles each day for 5 days. To find the total distance she ran, we need to multiply the distance she ran each day (2/3 miles) by the number of days she ran (5 days).

So, we can set up the multiplication like this:

Total distance Kayla ran = (2/3 miles) x (5 days)

To multiply fractions, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together.

So, the equation becomes:

Total distance Kayla ran = (2/3) x 5 miles

Multiplying 2/3 by 5 gives us:

Total distance Kayla ran = 10/3 miles

However, we usually want our answer to be in a simplified form. To simplify a fraction, we divide the numerator and denominator by their greatest common factor.

In this case, the greatest common factor of 10 and 3 is 1. So, our simplified answer is:

Total distance Kayla ran = 10/3 miles or 3 and 1/3 miles

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The length of a leg of the right triangle with a hypotenuse length of 10 cm is approximately 7 cm.

Applying the 45º-45º-90º Triangle Theorem, the length of a leg of a right triangle can be found by dividing the length of the hypotenuse by the square root of 2. In this case, with a hypotenuse length of 10 cm, the length of a leg can be determined.

The 45º-45º-90º Triangle Theorem states that in a right triangle with two equal legs, the length of a leg is equal to the length of the hypotenuse divided by the square root of 2.

In this case, the length of the hypotenuse is given as 10 cm. To find the length of a leg, we can divide the length of the hypotenuse by the square root of 2:

Leg = Hypotenuse / sqrt(2)

Substituting the given value, we have:

Leg = 10 cm / sqrt(2)

Using a calculator, the approximate value of the square root of 2 is 1.414. Therefore, we can calculate:

Leg = 10 cm / 1.414 ≈ 7.071 cm

Rounding to the nearest centimeter, the length of a leg is approximately 7 cm.

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The given linear programming problem can be analyzed by examining its constraints and objective function.

We are asked to minimize the objective function Z = 1X + 1Y, subject to the constraints:

1. 5X + 3Y < 30
2. 3X + 4Y > 36
3. Y < 7
4. X, Y > 0

To determine whether the problem exhibits infeasibility, unboundedness, or alternate optimal solutions, we'll analyze its feasible region.

Step 1: Plot the constraints on a graph and find the feasible region.
Step 2: Analyze the feasible region and identify its properties.

After plotting the constraints, we find that there is no common area satisfying all constraints.

This indicates that the problem exhibits infeasibility, meaning there is no solution that satisfies all the constraints simultaneously.

In this case, there are no alternate optimal solutions or unboundedness present since no feasible solution exists.

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To form a triangle, the set of line segments must meet the requirement that the sum of the lengths of any two sides is greater than the length of the third side. In other words, if we have three line segments, a, b, and c, to form a triangle, we need a + b > c, b + c > a, and a + c > b.

Therefore, we cannot determine the set of line segments that meet this requirement based on the information given in the question. We would need more information about the lengths of the line segments in order to determine which sets could form a triangle.

For example, if the set of line segments was {3, 4, 5}, then this set would meet the requirement to form a triangle because 3 + 4 > 5, 4 + 5 > 3, and 3 + 5 > 4. However, if the set of line segments was {1, 2, 6}, then this set would not meet the requirement to form a triangle because 1 + 2 < 6.

In summary, the set of line segments that meet the requirements to form a triangle depends on the lengths of the segments themselves, and more information is needed to answer this question.

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When it comes to printing full-color pictures, it usually involves using the CMYK color model which stands for cyan, magenta, yellow, and black.

These colors are combined in different amounts to create a wide range of colors and shades. Therefore, the answer to your question is three plus black. The colors cyan, magenta, and yellow are combined to produce a wide range of colors, while the black is added to enhance contrast and to create darker tones.

This combination of colors provides a more accurate and vibrant representation of the original picture. It is important to note that there are other color models used for printing such as RGB (red, green, and blue) which is used for digital displays, but for printing purposes, CMYK is the most common.

In conclusion, full-color pictures are printed using three colors (cyan, magenta, and yellow) plus black to enhance contrast and create darker tones. This combination of colors provides a more accurate and vibrant representation of the original picture.

This model consists of four ink colors: cyan (C), magenta (M), yellow (Y), and key (black, K). These four inks are combined in various proportions to produce a wide range of colors in the final print. Therefore, the correct answer would be four colors (CMYK), including black.

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Answer:

x^2 + y^2 = z^2

There are infinitely many choices for whole numbers x, y, and z.

By the pigeonhole principle, there must be at least two people at the party.

What is pigeonhole principle?

The pigeonhole principle is a fundamental concept in mathematics that states that if there are n items and k containers, and n > k, then at least one of the containers must contain more than one item.

This is an example of the pigeonhole principle. The pigeonhole principle states that if you have n pigeons and fewer than n pigeonholes, then there must be at least one pigeonhole with more than one pigeon in it.

In this case, we have 19 people and only 18 possible pigeonholes (since you cannot put two people in the same spot).

Therefore, by the pigeonhole principle, there must be at least one pigeonhole (i.e., a spot at the party) with more than one pigeon (i.e., more than one person). In other words, there must be at least two people at the party.

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The length of the arc ADC of the circle with center O is found to be 32.8 cm.

Since BA and BC are tangents to the circle, we know that OA and OC are perpendicular to BA and BC, respectively, and that they pass through the center of the circle O. Thus, triangle OAB and triangle OCB are right triangles, and we can use the Pythagorean theorem to find their sides,

OA = OB - AB = 12 - 7 = 5 cm

OC = OB - BC = 12 - 7 = 5 cm

Since A, C, and D are on the circle, we know that the length of arc ADC is equal to the length of the circumference of the circle between A and C, minus the length of the arc AB and BC,

arc ADC = (circumference of circle between A and C) - arc AB - arc BC

The circumference of a circle with radius 7 cm is,

C = 2πr = 2π(7) = 14π cm

The length of arc AB and BC are both equal to the length of the tangent segment from point B to the circle. Since BA and BC are both tangent to the circle, they are congruent, so we can just find the length of one of them,

AB = BC = √(OB² - OA²)

BC = √(12² - 5²)

BC = √119

Now we can substitute the values we found into the equation for the arc ADC,

arc ADC = (circumference of circle between A and C) - arc AB - arc BC

= 14π - √119 - √119

= 14π - 2√119

Using a calculator to evaluate this expression to 3 significant figures, we get,

arc ADC ≈ 32.8 cm

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Complete question - A, C and D are points on a circle of radius 7 cm, center O. BA and BC are tangents to the circle. OB = 12 cm. Work out the length of arc ADC. Give your answer correct to 3 significant figures.

In this scenario, a computer software magazine compares the rates of malware infection for computers protected by security software A and B.

To construct a 98% confidence interval for the difference between the proportions of infection for the two types of security software packages, follow these steps:

1. Calculate the proportions of infection for each security software (pA and pB):
  pA = 80/800 = 0.1
  pB = 45/900 = 0.05

2. Calculate the combined proportion (pC) and the standard error (SE) for the difference between proportions:
  pC = (80 + 45) / (800 + 900) = 125/1700 = 0.0735
  SE = sqrt((pC * (1 - pC)) * ((1/800) + (1/900))) = 0.0204

3. Find the z-score for a 98% confidence interval (z = 2.33 for a 98% CI).

4. Calculate the margin of error (ME) using the z-score and standard error:
  ME = z * SE = 2.33 * 0.0204 = 0.0475

5. Calculate the confidence interval for the difference between proportions:
  CI = (pA - pB) ± ME = (0.1 - 0.05) ± 0.0475 = 0.05 ± 0.0475 = (0.0025, 0.0975)

The 98% confidence interval is (0.0025, 0.0975), which can be rounded to (0.002, 0.098). This interval does not contain zero, which means it contradicts the claim that the proportion of infections is the same for the two types of security software. Therefore, the answer is (0.02, 0.098); yes.

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If the standard length of one kind of nail is 5 cm, and we want to test whether the nails produced on a particular day fit the standard requirement, we would set up the following hypotheses:

Null hypothesis (H0): The mean length of nails produced on the particular day is equal to the standard length of 5 cm.
Alternative hypothesis (Ha): The mean length of nails produced on the particular day is not equal to the standard length of 5 cm.

To test these hypotheses, we would take a sample of nails produced on the particular day and measure their lengths. We would then calculate the sample mean and compare it to the standard length of 5 cm using a hypothesis test.

In conclusion, If the sample mean is significantly different from the standard length, we would reject the null hypothesis and conclude that the nails produced on the particular day do not meet the standard length requirement. If the sample mean is not significantly different from the standard length, we would fail to reject the null hypothesis and conclude that the nails produced on the particular day meet the standard length requirement.

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The example that shows how changes in supply and demand is "where an employer has increased sales and needs to hire another person during lunch hour which increases income." Correct option is A.

In this case, the increased sales represent an increase in demand for the employer's product or service, which creates the need for additional labor.

As a result, the employer hires another person, which increases the number of workers and, subsequently, the supply of labor. However, since the demand for the product or service has increased, the price for labor also goes up, which leads to an increase in the income of the person who was hired.

This is an example of how the interaction between supply and demand can affect the price and quantity of goods and services, as well as the wages and salaries of workers.

Correct option is A.

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Answer:

As a lawn mower, I can earn $12 per hour without incurring any direct costs. However, my opportunity cost of time varies. For the first three hours, I could have earned $9 per hour doing other activities. Thereafter, my opportunity cost increases to $15 per hour due to other commitments. As such, my total earnings from lawn mowing depends on the number of hours I work, and I should prioritize lawn mowing in the first three hours to maximize my earnings.

the city grew by approximately 3 percent from 2010 to 2020.

To find out the percent growth from 2010 to 2020, we need to calculate the net percent change over the entire time period. We can start by finding out the population of the city in 2015.

From 2010 to 2015, the population grew by 6.6%. Using this percentage increase, we can find the population in 2015 by multiplying the 2010 population by 1.066:

179,000 x 1.066 = 190,294

Therefore, the population in 2015 was approximately 190,294.

From 2015 to 2020, the population fell by 3.4%. Using this percentage decrease, we can find the population in 2020 by multiplying the 2015 population by 0.966:

190,294 x 0.966 = 183,902

Therefore, the population in 2020 was approximately 183,902.

To find the net percent change from 2010 to 2020, we can use the formula:

[(final value - initial value) / initial value] x 100

Plugging in the numbers we found, we get:

[(183,902 - 179,000) / 179,000] x 100 = 2.7%

Therefore, to the nearest whole number, the city grew by approximately 3% from 2010 to 2020.

The city experienced a period of growth from 2010 to 2015, followed by a period of decline from 2015 to 2020. However, overall, the city still managed to grow by approximately 3% over the entire time period.


1. Calculate the population in 2015 by applying the 6.6% growth:
2015 Population = 179,000 * (1 + 6.6/100) = 179,000 * 1.066 ≈ 190,814

2. Calculate the population in 2020 by applying the 3.4% decline:
2020 Population = 190,814 * (1 - 3.4/100) = 190,814 * 0.966 ≈ 184,302

3. Calculate the overall percentage growth from 2010 to 2020:
Percentage Growth = ((2020 Population - 2010 Population) / 2010 Population) * 100
Percentage Growth = ((184,302 - 179,000) / 179,000) * 100 ≈ 2.96%


Considering the population growth and decline in the respective periods, the city's population grew by approximately 3% from 2010 to 2020, to the nearest whole number.

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Answer:

[tex] \frac{1}{.41} = 2.4390[/tex]

One British pound should sell for $2.4390.

D is correct.

In a soccer league the probability of playing against an in-state team is "0.6".

Given details:

Total no. of teams = 10

Total no. of in- state teams = 6

Total no. of out of  state teams = 4.

The probability of playing against an in - state team is :

Probability = ( Number of favourable cases / Total outcomes).

= 6/10

= 0.6.

Therefore, the probability of playing against an in-state team is "0.6".

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The total amount of interest earned by the investment in n years is 100 times the product of the annual interest rate and the time period i.e., I = 100 * x * n

The total amount of interest earned by an investment can be calculated using the simple interest formula:

I = P * r * t

Where:

I = the total amount of interest earned

P = the principal investment amount

r = the annual interest rate as a decimal

t = the time period in years

In this case, the principal investment amount is $10,000, the annual interest rate is x percent (as a decimal, x/100), and the time period is n years. So, the total amount of interest earned can be represented by:

I = 10,000 * (x/100) * n

Simplifying this expression, we get:

I = 100 * x * n

Therefore, the total amount of interest earned by the investment in n years is 100 times the product of the annual interest rate and the time period. The units of this value will be dollars, since it represents the amount of interest earned.

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the approximation of the number of people who leave the building during the first 15 minutes of the evacuation drill using a right Riemann sum with three subintervals is 375 people.

To approximate the number of people who leave the building during the first 15 minutes of the evacuation drill using a right Riemann sum with three subintervals, we can divide the interval [0, 15] into three subintervals of equal width:

[0, 5], [5, 10], [10, 15]

The right Riemann sum is then given by:

Δt [f(5) + f(10) + f(15)]

where Δt is the width of each subinterval (Δt = 5), f(t) is the rate of people leaving the building at time t (in people per minute), and the values of f(t) are given in the table.

Plugging in the values, we get:

Δt [f(5) + f(10) + f(15)]

= 5 [20 + 25 + 30]

= 5 [75]

= 375

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The given polynomial has seven terms.

It should be noted that to count the number of terms in a polynomial, we need to identify each individual term, which are separated by plus or minus signs.

In this case, we can write the polynomial as:

-10x^6 - 2x^5 + 7x^3 - 9x^2 + 4x^1 + 5.5x^1 - 1

The individual terms are:

-10x^6, -2x^5, 7x^3, -9x^2, 4x^1, 5.5x^1, -1

There are seven terms in total.

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Answer:

sin 40=280/diagonal,diagonal=280/sin40=435.6

x=cos40×435.6=333.68

A=333.68×280=93,423.9m^2

The value of the given expression is 3.75.

Given is an expression, 3/2·∛16, we need to solve it,

3/2·∛16
= 1.5 × ∛16

= 1.5 × ∛2×2×2×2

= 1.5 × 2∛2

= 1.5 × 2 × 1.25

= 3.75

Hence, the value of the given expression is 3.75.

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Under the null hypothesis. Using a standard normal table or a calculator, we find that the P-value is between 0.05 and 0.10.

The answer is c) between 0.05 and 0.10.

In statistics, the null hypothesis (H0) is a statement that assumes that there is no significant difference between two or more groups, samples, or populations.

To test for the equality of the proportions of streams that fail to meet minimum requirements in the two areas, we can use a two-sample test of proportions.

Let p1 be the true proportion of streams that fail to meet minimum requirements in group 1 (Eastern Virginia), and p2 be the true proportion of streams that fail to meet minimum requirements in group 2 (Western Virginia). The null hypothesis is that the two proportions are equal, i.e., H0: p1 = p2, and the alternative hypothesis is that they are not equal, i.e., Ha: p1 ≠ p2.

We can use the pooled estimate of the proportion, p, to test the null hypothesis. The formula for the pooled estimate of the proportion is:

p = (x1 + x2) / (n1 + n2)

where x1 and x2 are the numbers of streams that fail to meet minimum requirements in groups 1 and 2, respectively, and n1 and n2 are the sample sizes.

The test statistic is:

z = (p1 - p2) / √(p * (1 - p) * (1/n1 + 1/n2))

Under the null hypothesis, the test statistic follows a standard normal distribution.

The observed values are x1 = 17, n1 = 85, x2 = 24, n2 = 80.

The pooled estimate of the proportion is:

p = (17 + 24) / (85 + 80) = 0.202

The test statistic is:

z = (17/85 - 24/80) / √(0.202 * (1 - 0.202) * (1/85 + 1/80)) = -1.78

The P-value for a two-sided test is the probability of observing a test statistic as extreme as -1.78 or more extreme, under the null hypothesis. Using a standard normal table or a calculator, we find that the P-value is between 0.05 and 0.10.

Therefore, the answer is c) between 0.05 and 0.10.

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The answer is option B) 09. To find the mean (average) of the yards gained by Roger during his seven carries, we need to add up all the yards gained and divide the sum by the number of carries.

So, for the given data set {2, 6, 20, 11, 8, 12, 4}, the total yards gained is:

2 + 6 + 20 + 11 + 8 + 12 + 4 = 63

The number of carries is 7.

Therefore, the mean yards gained per carry is:

Mean = Total yards gained / Number of carries

= 63 / 7

= 9

Hence, the answer is option B) 09.

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Answer:

Step-by-step explanation:

3x+4+x=16

Combine like terms

(3x+x)+4=16

subtract 4

4x=12

Divide by 4

x=3

The length of the shorter side is approximately 23.66 inches, and the length of the longer side is twice as long, or approximately 47.32 inches.

To find the length of the longest side, which is the hypotenuse, we're also given that it's 30 inches long.

Now, let's use what we know about the relationships between the sides of a 30-60 degree right triangle to find the length of the other two sides.

First, we know that the side opposite the 60 degree angle is always twice as long as the side opposite the 30 degree angle. So let's call the length of the shorter side (opposite the 30 degree angle) "x". Then the length of the longer side (opposite the 60 degree angle) is 2x.

Next, we can use the Pythagorean theorem to relate the lengths of the sides. In a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. So we have:

30² = x² + (2x)²

Simplifying this equation, we get:

900 = 5x²

Dividing both sides by 5, we get:

x² = 180

Taking the square root of both sides, we get:

x = √180

Now, we know that the perimeter of the triangle is the sum of the lengths of all three sides. So we have:

70.98 = x + 2x + 30

Simplifying this equation, we get:

100.98 = 3x + 30

Subtracting 30 from both sides, we get:

70.98 = 3x

Dividing both sides by 3, we get:

x = 23.66

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Answer:

The best way to decompose the composite figure to determine its area is as a semicircle, a trapezoid, and two rectangles. This way, we can use the following formulas to find the area of each part:

Area of a semicircle = 21​πr2, where r is the radius of the circle.

Area of a trapezoid = 21​(b1​+b2​)h, where b1​ and b2​ are the bases and h is the height of the trapezoid.

Area of a rectangle = l×w, where l is the length and w is the width of the rectangle.

Then, we can add up the areas of each part to find the total area of the composite figure. The other options are not as convenient because they either involve more parts or more complicated shapes. For example, option A would require finding the area of a triangle, which involves using trigonometry or the Pythagorean theorem. Option B would require finding the area of four triangles, which is more tedious than finding the area of two rectangles. Option C would require finding the area of a square, which is redundant because a square is a special case of a rectangle.

Step-by-step explanation:

The composite figure can be decomposed into a semicircle, a trapezoid, and two rectangles.

Option D is the correct answer.

It is a quadrilateral that has one pair of parallel sides and a height.

The area is calculated as: 1/2 x sum of the parallel sides x height.

Examples:

Area of a trapezium that has the parallel sides as 3 cm and 4 cm and a heght o 5 cm.

Area = 1/2 x (3 + 4) x 5

Area = 1/2 x 7 x 5

Area = 35/2 = 17.5 cm^2

We have,

The given figure can be decomposed into the following figure.

- Semicircle

- Trapezoid

- Two rectangles

Thus,

A semicircle, a trapezoid, and two rectangles.

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