4. Astronomers have discovered a new constellation made up of Star A, Star B, and Star C that forms
a right triangle. The angle at Star A is 90 degrees. Star A and Star B are 50 lightyears apart, while
Star A and Star C are 120 lightyears apart. How many lightyears apart are Star B and Star C?

The value of r is greater than 1. This tells that the series diverges. The solution has been obtained by using ratio test.

What is ratio test?

One test used to assess if infinite series are converging or diverging is the ratio test. Even finding the radius and interval of power series convergence using the ratio test is an option.

We are given an infinite series as

x(n) = 2n! / 2²ⁿ

Using the ratio test,

[tex]\lim_{n \to \infty} \frac{\frac{2(n+1)!}{2^{2(n+1)} } }{\frac{2n!}{2^{2n} } }[/tex]

[tex]\lim_{n \to \infty} \frac{n+1}{4}[/tex] = ∞ = r

So, we observed that the value of r is greater than 1.

We know that if r is greater than 1, then the series diverges.

So, the given series also diverges.

Hence, the value of r is greater than 1 which shows that the series diverges.

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(a) The number of washers that can be cut from a sheet of steel is 12,732 washers.

(b) No additional washers could be cut from the metal left over after it was remelted and cast into a sheet of the same thickness and width (1 m).

First, we need to calculate the area of the washer, which can be done as follows:

Area = π (de/2)² - π (di/2)²

= π(0.02 m/2)² - π (0.01 m/2)²

= 0.000236 m²

Now, we can calculate the number of washers that can be cut from a 3m x 1m sheet of steel:

Number of washers = (Area of the sheet) / (Area of the washer)

= (3 m x  1 m) / 0.000236 m²

= 12,732 washers

Next, we need to find the area of the metal left over after cutting the 12,732 washers. This can be calculated as follows:

Area of the metal left over = (3 m²) - (12,732  x 0.000236 m²)

= 3 - 3.05

= -0.05 m²

Since the metal left over is negative, it means that not enough metal was available to cut 12,732washers. Hence, 12,732 is the maximum number of washers that can be cut from a 3m x 1m sheet of steel.

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Since the research company is making a claim about the proportion of college students who shop online and Jamila is taking a sample of college students to test this claim, she should use a one-proportion z-test.

A one-proportion z-test is used to test whether the proportion of successes in a sample is significantly different from a hypothesized proportion. Here, the hypothesized proportion is 75% (the claim made by the research company) and the sample proportion is 120/200 = 0.6.

Jamila can use the one-proportion z-test to determine whether the difference between the sample proportion and the hypothesized proportion is statistically significant. The null hypothesis is that the sample proportion is equal to the hypothesized proportion (i.e., p = 0.75), and the alternative hypothesis is that the sample proportion is not equal to the hypothesized proportion (i.e., p ≠ 0.75).

Based on the sample size and the fact that the population standard deviation is not known, Jamila should use the normal approximation to the binomial distribution to perform the one-proportion z-test.

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Answer:

Step-by-step explanation:

Find the GCF:

12, 24, 36

8, 16, 24, 32

GCF is 24

The lowest amount of boxes the teacher could buy would be 2 boxes of Rulers and 3 boxes of Protractors. I hope this helped

1. The function for this problem is defined as follows: y = 2x + 3.

2. The numeric value of the function at x = 2 is given as follows: y = 7.

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

We define a function as follows:

y = 2x + 3.

At x = 2, the numeric value is found replacing the lone instance of x by 2, hence:

y = 2(2) + 3

y = 7.

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Answer:

  Two solutions:

Step-by-step explanation:

Given right ΔABC, with tan(A) = 3/2 and b = 5, you want the exact lengths of sides a and c.

To completely solve a triangle, you need to know at least one side, at least one angle, and one other side or angle. Here, we're given one side and one angle, so cannot solve the triangle using the given information.

We know another angle is 90°, but we don't know which one.

Often, side c is the one designated as the hypotenuse, but that is not necessarily the case. So, there are two solutions.

This case corresponds to the red triangle shown in the attachment.

In this triangle, the leg adjacent to angle A is side b, so we have ...

  Tan = Opposite/Adjacent . . . . . trig relation

  3/2 = a/5

  a = 15/2

The Pythagorean theorem can be used to find the length of side c.

  c² = a² +b²

  c² = (15/2)² +5² = 225/4 +25 = 325/4

  c = √(325/4)

  c = (5/2)√13

This case corresponds to the blue triangle shown in the attachment.

The given tangent of angle A tells us the ratio ...

  Tan = Opposite/Adjacent

  3/2 = a/c

  a = 3/2c . . . . . multiply by c

The Pythagorean theorem tells us ...

  b² = a² +c²

  5² = (3/2c)² +c² = 13/4c² . . . . . . substitute for b and a

  c² = (4/13)(5²) = (10/13)²·13 . . . . . . multiply by 13/13

  c = (10√13)/13 . . . . . . . . . . . . square root

  a = 3/2c = (15√13)/13 . . . . . . find 'a'

The expression that represents the amount of money the friends still need to collect to meet their goal is 5x² - 7xy + 20. The solution has been obtained by using arithmetic operations.

What are arithmetic operations?

The four fundamental mathematical operations that can be used to express any real number are referred to as "arithmetic operations" in mathematics. The four operations are addition, subtraction, division, and multiplication.

We are given the total contribution goal as 8x² - 4xy + 9

Amounts collected by:

Martin = 3xy + 17

Tyreese =  x²

Braydon = 2x² - 6

So, the total amount collected till now is

(3xy + 17) + x² + (2x² - 6) = 3x² + 3xy - 11

Amount left to be collected is

⇒(8x² - 4xy + 9) - (3x² + 3xy - 11)

⇒8x² - 4xy + 9 - 3x² - 3xy + 11

⇒5x² - 7xy + 20

Hence, the expression is 5x² - 7xy + 20.

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The number of cans collected by Sasha to reach her goal is equal to 300 cans.

Number of cans collected by Sasha before reaching her final goal of recycling is equal to 180 cans.

Percent of cans collected by which is equal to 180 is 60%

Let 'y' be the total number of cans collected by Sasha to reach her goal .

60% of y = 180

⇒ ( 60 / 100 ) × y = 180

⇒ y = ( 180 × 100 ) / 60

⇒ y = 18000 / 60

⇒ y = 300 cans

Therefore, the total number of cans collected by Sasha to reach her goal of recycling is equal to 300 cans.

The above question is incomplete, the complete question is :

Sasha is collecting cans to recycle. When she collected 180 cans she knew she had reached 60% of her goal.

Question: How many more cans does Sasha need to collect to reach her goal?

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68.5% of Colton's gross pay is take-home pay. To find the percentage of Colton's gross pay that is take-home pay, we need to divide his net pay by his gross pay and then multiply by 100 to convert the resulting decimal to a percentage.

To find the percentage of Colton's gross pay that is take-home pay, we need to divide his net pay by his gross pay and multiply the resulting decimal by 100 to express it as a percentage. Colton's gross pay is $1421.50 and his net pay is $973.48. Dividing the net pay by the gross pay gives a result of 0.685, or 68.5% when converted to a percentage. This means that 68.5% of Colton's gross pay is take-home pay, or the amount of money that he receives after deductions are made. Therefore, Colton's take-home pay is $973.48, which is 68.5% of his gross pay.

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In response to the given problem, we can state that the fraction sum is a rational fraction.  The numerators of true fractions are smaller than the denominators.

To represent a whole, any number of equal parts or fractions can be utilized. In standard English, fractions show how many units there are of a particular size. 8, 3/4. Fractions are part of an entire. In mathematics, numbers are stated as a ratio of the numerator to the denominator.

They can all be expressed as simple fractions as integers. A fraction appears in a complex fraction's numerator or denominator.

A sum that is a fraction of a total is called a fraction. You can analyse something by dissecting it into smaller pieces. For instance, the number 12 is used to symbolize half of a full number or object.

And

[tex]-11/4 + 5/9[/tex]

[tex]-99+20 / 36 = -79/36[/tex]

Therefore,  By fraction, sum is fraction as rational.

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The required probability of loosing both the contracts with 61% chances of winning first contract and 74% chances of second contract is equal to 0.1014.

Probability of winning the first contract 'P(A)' = 61%

                                                                  = 0.61

Probability of winning the second contract 'P(B)' = 74%

                                                                       = 0.74

Probability of loosing both the contracts is

=  [ 1 - P(A) ] × [ 1 - P(B) ]

= [ 1 - 0.61 ]  × [ 1 - 0.74 ]

= 0.39 × 0.26

= 0.1014

Therefore, the probability of loosing both the contracts as per given contracts winning possibility is equal to 0.1014.

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The population of Toliver town reached a maximum of 200 by the mid - 1890s.

After the Civil War, people initially began settling in Toliver, a farming community two miles northwest of the Cushing site in northwestern Nacogdoches County. There was a post office created there in 1888, and by the middle of the 1890s, there were 200 people living there.

However, the community was bypassed when the Texas and New Orleans Railroad was built through the region in 1901, and the majority of its population and businesses went to the recently founded town of Cushing, on the rail line, and the town was abandoned a few years later.

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The full question is:

What was the population of Toliver town​ at it's height ?

Answer: 1/9 is the fraction form

Step-by-step explanation:

Answer:

1/9

Step-by-step explanation:

[tex]3^{-2}[/tex]

= 1/[tex]3^{2}[/tex]

= 1/9

The area of the walkway of the garden is  8x - 15 square units.

The area of a rectangle is the product of its length and width.

The perimeter of a rectangle is the sum of the lengths of all the sides.

Given, The area of the garden is 2(3.5x + 2.5) and the combined area of the garden and the walkway is 2.5(6x + 4).

Therefore, The area of the walkway is,

= 2.5(6x + 4) - 2(3.5x + 2.5).

= 15x - 10 - 7x - 5.

= 15x - 7x - 10 - 5.

= 8x - 15 square units.

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The double-angle identity for cosine is cos(2x) = cos²(x) - sin²(x).

To derive the reduction identities, we can use this double-angle identity and manipulate it to express cosine or sine of x in terms of cosine or sine of a smaller angle. Here are the steps for deriving the reduction identity for cosine: Start with the double-angle identity: cos(2x) = cos²(x) - sin²(x).

Divide both sides by cos²(x): cos(2x) / cos²(x) = (cos²(x) - sin²(x)) / cos²(x).

Use the identity sin²(x) + cos²(x) = 1 to replace cos²(x) in the denominator: cos(2x) / cos²(x) = (1 - sin²(x)) / cos²(x).

Rearrange the right-hand side: cos(2x) / cos²(x) = 1/cos²(x) - sin²(x) / cos²(x).

Recognize that 1/cos²(x) is equal to sec²(x), and use the identity tan²(x) + 1 = sec²(x) to replace it: cos(2x) / cos²(x) = sec²(x) - tan²(x).

Simplify using the identity cos(x)/sin(x) = cot(x): cos(2x) / cos²(x) = cosec²(x) - cot²(x).

Simplify the left-hand side using the identity cos(2x) = 2cos²(x) - 1: (2cos²(x) - 1) / cos²(x) = cosec²(x) - cot²(x).

Rearrange and simplify: 2 - (1/cos²(x)) = cosec²(x) - cot²(x).

Recognize that 1/cos²(x) is equal to sec²(x) and cosec(x) is equal to 1/sin(x): 2 - sec²(x) = 1/sin²(x) - cot²(x).

Simplify the left-hand side using the identity 1 - sin²(x) = cos²(x): cos²(x) - sec²(x) = 1/sin²(x) - cot²(x).

Recognize that sec(x) is equal to 1/cos(x) and cot(x) is equal to cos(x)/sin(x): cos²(x) - 1/cos²(x) = sin²(x)/cos²²(x) - cos²x)/sin²(x).

Simplify: cos⁴(x) - 1 = sin²(x) - cos⁴(x).

Rearrange: 2cos⁴(x) = sin²(x) + 1.

Finally, recognize that sin²(x) + cos²(x) = 1, and use it to replace sin²(x): 2cos⁴(x) = cos²(x). This is the reduction identity for cosine: cos²(x/2) = (1 + cos(x))/2.

To derive the reduction identity for sine, we can use the same double-angle identity for cosine and manipulate it to express sine of x in terms of sine or cosine of a smaller angle. The steps are similar but may require using different.

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The percentage of the increase in the temperature of the mixture is 27.34%.

What is the percentage?

A percentage is a figure or ratio that may be stated as a fraction of 100 in mathematics. If we need to compute the percentage of a number, divide it by the entire and multiply by 100. As a result, the percentage denotes a part per hundred. Per 100 is what the term percent signifies. The sign "%" represents it.

Here are some percentage examples:

10% is 1/10 of a fraction.

20% is equal to a 1/5 portion.

25% is equal to 1/4 fractions.

Now,

As percentage increase = (final value-initial value) / initial value *100

Here initial temperature=12 1/4 F=49/4 F and

Final temperature = 15 3/5=78/5 F

then (final value-initial value) =78/5-49/4=15.6-12.25

=3.35

therefore, percentage increase=3.35/12.25*100

=0.2734*100

=27.34%

Hence,

           The percentage of the increase in the temperature of the mixture is 27.34%.

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The square root of 378 is 19.4422

The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol used to denote the square root is √, and it is placed in front of the number for which we are looking for the square root.

First, let's define what factorial means. The factorial of a non-negative integer is the product of all positive integers less than or equal to that number.

Next, we have to compute the square root of 378, which is a much simpler task. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 x 5 = 25. Using a calculator, we can find that the square root of 378 is approximately 19.4422.

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Complete Question:

What is the square root of 378?

1) Note that the volume of water that was already in the pool when the hose was turned on today is 2000 gallons of water.

2) The equation that represents the relationship between the volume of water in the pool in gallons "V) and the elapsed time in hours, t, since the hose was turned on today is V = 800t + 2000.

Note that to get the right expression above we have to solve for the slope of the graph.

1) To arrive at the answer in 1 above, we only have to examine the y-intercept. since at time (t) =0 the volume of water (gallons) indicated on the graph is 2000 gallons, thus, it is correct to state that:

"the volume of water that was already in the pool when the hose was turned on today is 2000 gallons of water."

2) From the graph, we can see the equation must be linear.
That is:

y = mx + b

Where y = V = Volume of Water in the Pool in Gallons

m = Slope

x = t = elapsed time in hours
b = y- intercept = already given as 2000

Thus:

To get m (slope) we need to use the above equation:
m = rise/run = y2-y1/x2-x1

Take
y1 = 6000

y2 = 10,000

x1 = 5

x2 = 10

Substituting the given values, we get:

m = (10,000 - 6,000) / (10 - 5)

m = 4,000 / 5

m = 800

Therefore, the slope of the line passing through the points (5, 6000) and (10, 10,000) is 800.
Having derived m, the full expression is thus given as:

V = 800t + 2000.

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Full Question:
See attached for full question

The equation of a circle whose center is (0,.-5) and radius is 7 as required is; (x - 0)² + (y + 5)² = 7².

It follows from circle equations that; (x - a)² + (y - b)² = r² represents the equation of a circle whose center is; (a, b) and radius is; r.

Hence, for the given circle whose center is (0,.-5) and radius is 7.

(x - 0)² + (y + 5)² = 7²

Therefore, it can be inferred that the equation of the circle in discuss is; (x - 0)² + (y + 5)² = 7².

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The actual dimensions of the figure is 3.9 x 2.1

Scale factor is a mathematical term, which we use to scale 2-dimensional and 3-dimensional shapes. Scale factor is a measure of similar figures. Those figures who look the same but are in different sizes.

For example, Two spheres look similar but could have different radii.

For the first dimension of 7.8 cm, we can set up a proportion using the scale factor:

2 cm : 1 m = 7.8 cm : x

where x is the actual length we want to find.

Solving for x, we get:

x = (7.8 cm x 1 m) / 2 cm

x = 3.9 m

Therefore, the first actual dimension is 3.9 meters.

For the second dimension of 4.2 cm, we can use the same method:

2 cm : 1 m = 4.2 cm : y

where y is the actual width we want to find. Solving for y, we get:

y = (4.2 cm x 1 m) / 2 cm

y = 2.1 m

Therefore, the second actual dimension is 2.1 meters.

So the actual dimensions of the figure are 3.9 meters by 2.1 meters.

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Answer: If the transformation of the parent function f(x) = √x is a reflection over the x-axis, it means that the graph of the transformed function g(x) is obtained by reflecting the graph of f(x) over the x-axis.

The reflection of the function f(x) over the x-axis is obtained by negating its output, so the function g(x) can be written as:

g(x) = -√x

So the transformed function g(x) is a reflection of the parent function f(x) = √x over the x-axis.

Step-by-step explanation:

Step-by-step explanation:

oh, come on ! we are dealing with rectangles.

and you know how to get the perimeter and the area of a rectangle. upper elementary school students can do that.

remember, the perimeter is the distance when you walk all around the rectangle.

so, 2 times the length and 2 times the width.

and the area is simply length × width.

but your teacher was very lazy and careless with some phrasing about what is needed.

the original room is 8 ft × 3 ft.

1.

the length of border needed is exactly the perimeter.

so,

2×8 + 2×3 = 16 + 6 = 22 ft

2.

the area of the floor is

8×3 = 24 ft²

this is the amount of floor panels to be bought. we buy them in ft².

3.

the new room is (8+3) ft × (3-2) ft. = 11 ft × 1 ft.

that is a very strange room. no human can live in such a room.

I think your teacher meant (8-2) ft × (3+3) ft = 6 ft × 6 ft.

but let's stick with the original information.

the border for this room (perimeter) is

2×11 + 2×1 = 22 + 2 = 24 ft

4.

the floor area is

11 × 1 = 11 ft²

if we consider my correction of the room size, it is

6×6 = 36 ft²

5.

"difference in size" is not clear. regarding the dimensions (side lengths and correspondingly the perimeter) or regarding the floor size ?

if we consider the perimeter, the difference is

24 - 22 = 2 ft

the new room's perimeter is 2 ft less.

if we consider the floor area, the difference is

24 - 11 = 13 ft²

the new room is 13 ft² smaller.

or for my size correction, the difference is

36 - 24 = 12 ft²

the new room is then 12 ft² larger.

It is a biased sample.

What is meant by a random sample?

Each sample has an equal chance of being chosen as part of the sampling procedure known as random sampling. A randomly selected sample is intended to be a fair reflection of the entire population.

A random sample is one in which each individual, thing, or event has an equal chance of being chosen. Compared to other types of samples, a random sample has a higher likelihood of being representative of the total population. A skewed sample is one that does not fairly represent the population as a whole.

Here Dorthy considered only the first 20 people and not the whole population so this is not a random sample and it is a biased sample.

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The correct answer is "the more accurately the scores are represented by the mean."

The variability or dispersion of a set of scores refers to how spread out the scores are from the central tendency, which is usually measured by the mean, median, or mode.

When a set of scores has low variability or dispersion, it means that the scores are clustered closely together around the mean, and therefore, the mean is a more accurate representation of the typical score in the set.

On the other hand, when a set of scores has high variability or dispersion, it means that the scores are more spread out from the mean, and therefore, the mean may not be a very accurate representation of the typical score in the set. In such cases, other measures of central tendency, such as the median or mode, may provide a more accurate representation of the scores.

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Cases are on display OA 619,200 cubic inches.

Number is a mathematical entity used to represent a computer magnitude it can be symbol or a combination of simple you should in order to take quantity such as the two, five, seven, three or eight number are used to verify the context including counting measuring and computing.
To calculate the number of cubic inches needed for 360 cases of 12-ounce cans of soft drink, we need to multiply the number of cases (360) by the volume of each case. The volume of each case can be calculated by multiplying its length (16 inches), width (10.75 inches) and height (5 inches):

Volume = 16.00 inches x 10.75 inches x 5.00 inches = 828 cubic inches

Therefore, 360 cases of 12-ounce cans of soft drink require (360 x 828) = 619,200 cubic inches.

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The sample mean is 89.45, the sample standard deviation is 2.327, and the sample median is 91 and the leaf plot of the data is illustrated below.

The sample standard deviation is a measure of how much the ratings vary from the sample mean. It tells us how spread out the data is. A larger standard deviation means the data is more spread out, while a smaller standard deviation means the data is more tightly clustered around the mean.

To calculate the sample standard deviation, we first need to calculate the variance. The variance is the average of the squared differences between each rating and the mean. The formula for variance is:

variance = (sum of (rating - mean)²) / (number of ratings - 1)

In this case, the sum of the squared differences is 301.9, and there are 39 degrees of freedom (40 ratings minus 1), so the variance is 7.741. To get the sample standard deviation, we take the square root of the variance. In this case, the sample standard deviation is approximately 2.78.

The sample median is the middle rating when the ratings are arranged in order from lowest to highest. In this case, there are 40 ratings, so the median is the average of the 20th and 21st ratings, which are both 91.

Finally, we can determine what proportion of the taste-tasters considered this particular pinot noir truly exceptional (above 90 points). To do this, we count the number of ratings that are above 90 and divide by the total number of ratings. In this case, there are 30 ratings above 90, so the proportion is 30/40, or 0.75. This means that 75% of the taste-tasters considered this pinot noir truly exceptional.

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Complete Question:

A group of wine enthusiasts taste-tested a pinot noir wine from Oregon. The evaluation was to grade the wine on a 0-to-100-point scale. The results follow. Construct a stemand- leaf diagram for these data and comment on any important features that you notice. Compute the sample mean, the sample standard deviation, and the sample median. A wine rated above 90 is considered truly exceptional. What proportion of the taste-tasters considered this particular pinot noir truly exceptional?

94 90 92 91 91 86 89 91 91 90

90 93 87 90 91 92 89 86 89 90

88 95 91 88 89 92 87 89 95 92

85 91 85 89 88 84 85 90 90 83

Answer:

Step-by-step explanation:

I just did this q. today at school but Imn't in a mood to write whole steps but.
x=4 and QPR= 102

the equation I used:
[tex]26x-2=\frac{1}{2} (131+18x+1)[/tex]

The required value of y is 8/12 and the sequence is given as 1/2, 7/12, 8/12, 3/4.

Arithmetic progression is the series of numbers that have common differences between adjacent values

Here,
Let the common difference between the values be d = 1/12
Now

Since x in the second term
x = 1/2 + (2- 1 )1/12
x = 7/12

Since y is the third term
So,
y = 1/2 + (3 - 1)d
y = 1/2 + 2/12
y = 1/2 + 1/6
y = 6 + 2/12
y = 8/12

Thus, the required value of y is 8/12 and the sequence is given as 1/2, 7/12, 8/12, 3/4.

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Answer:

1. 285 centiliters

2. yards-->inches-->centimeters

Step-by-step explanation:

Step-by-step explanation:

First things first, we are rotating about a point so the first we do is the move the clockwise rotation to the origin.

Since a rotation is a rigid transformation, we must move all the other points as well.

So move every point using the rotation vector

R(-6,-8), move every point to

D(1-6,4-8)=D'(-5,-4)

E(5-6,5-8)=E'(-3,-5)

F(4-6,1-8)=F'(-2,-7)

G(0-6,0-8)=G'(-6,-8)

So know since the center of rotation is middle at (0,0) , we can apply the rotation rules

For a 270 clockwise rotation, we apply this rule

(-y,x)

So for the points we know get

D''(5,-4)

E''(3,-5)

F''(2,-7)

G"(6,-8)

Know since we done our rotation, we undo what we did in step 1, we add 6 to the x coordinates and 8nto the y coordinate

D"'(11,8)

E"'(9, 3)

F"(8,1)

G"'(12,0)

So those are our new coordinates.