Mathematics College

A rectangle is placed around a semicircle as shown below. The width of the rectangle is . Find the area of the shaded region.Use the value for , and do not round your answer. Be sure to include the correct unit in your answer.

Answers

Answer 1

Solution

Step 1

Write the given data:

Radius r of the semi-circle = 4 yd

Width of the rectanhle = 4 yd

Length of the rectangle = 2 x 4 = 8 yd

Step 2

Write the formula for the area of the shaded region:

[tex]\begin{gathered} Area\text{ of the shaded region} \\ =\text{ Area of a rectangle - Area of the semi-circl} \\ =\text{ W }\times\text{ L - }\frac{\pi r^2}{2} \\ =\text{ 4}\times\text{ 8 - }\frac{3.14\times4^2}{2} \\ =\text{ 32 - 25.12} \\ =\text{ 6.88 yd}^2 \end{gathered}[/tex]

Final answer

6.88

Related Questions

Which of the following numbers are greater than 6 and less than 8? Explainhow you know.

Answers

Let's analyze each case and see if they are less than 8 first:

[tex]7<8,therefore(7)^{\frac{1}{2}}<8[/tex][tex]\sqrt[]{64}=8[/tex]

thus

[tex]\sqrt[]{60}<\sqrt[]{64}=8[/tex][tex]\sqrt[]{60}<8[/tex]

Finally,

[tex]64<80[/tex][tex]\sqrt[]{64}<\sqrt[]{80}[/tex]

thus

[tex]8<\sqrt[]{80}[/tex]

In sumary, the first two options are less than 8, but not the third.

Now let's see if they are greater than 6

[tex]9>7[/tex][tex]\sqrt[]{9}>\sqrt[]{7}[/tex][tex]3>\sqrt[]{7}[/tex]

and

[tex]6>3>\sqrt[]{7}[/tex]

thus

[tex]6>\sqrt[]{7}[/tex]

Now

[tex]36<60[/tex][tex]\sqrt[]{36}<\sqrt[]{60}[/tex]

and so

[tex]6<\sqrt[]{60}[/tex]

Finally

[tex]\sqrt[]{80}>\sqrt[]{60}>6[/tex]

thus

[tex]\sqrt[]{80}>6[/tex]

In conclussion, the second and third options are greater than 6, but not the first.

the discriminant equation How many real solution 4x^2-8x+10=-x^2-5 have?

Answers

Answer:

0 real solutions

Explanation:

First, we need to transform the equation into the form:

[tex]ax^2+bx+c=0[/tex]

So, the initial equation is equivalent to:

[tex]\begin{gathered} 4x^2-8x+10=-x^2-5 \\ 4x^2-8x+10+x^2+5=-x^2-5+x^2+5 \\ 5x^2-8x+15=0 \end{gathered}[/tex]

Now, the discriminant can be calculated as:

[tex]b^2-4ac[/tex]

If the discriminant is greater than 0, the equation has 2 real solutions.

If the discriminant is equal to 0, the equation has 1 real solution

If the discriminant is less than 0, the equation has 0 real solutions

So, in this case, a is 5, b is -8 and c is 15. Then, the discriminant is equal to:

[tex](-8)^2-4\cdot5\cdot15=84-300=-236[/tex]

Since the discriminant is less than zero, the equation has 0 real solutions

in the experiment of the preceding exercise, the subjects were randomly assigned to the different treatments. what is the most important reason for this random assignment?

Answers

The most important reason for random assignment on the subjects in the experiment, is because random assignment would be the best way in creating group of subjects to the different treatments.

Note that; the group of subjects are roughly equivalent at the beginning of the experiment.

Using random assignment will allow the allocation of different patients to various treatments at a random order. From this there will be objective results obtained altogether from the experiment under investigation.

Random assignment will eliminate any biasness that may occur when conducting the experiment. It prevents favoritism of any event from occurring. It will ensure that all the different patients have an equal chance of being selected for various treatment.

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what is the sum of -1 1/3 + 3/4

Answers

[tex]-1\frac{1}{3}+\frac{3}{4}\text{ = -}\frac{7}{12}[/tex]

Here, we want to add two fractions

What we have to do here is to make the mixed fractin an improper one

To do this, we multiply the denominator by the standing number, and add to the numerator, then we place the value over the denominator

Thus, we have it that;

[tex]\begin{gathered} 1\frac{1}{3}\text{ = }\frac{4}{3} \\ -\frac{4}{3}+\frac{3}{4}\text{ = }\frac{-16+9}{12}=\text{ }\frac{-7}{12} \end{gathered}[/tex]

A bag contains 8 red marbles, 7 blue marbles and 6 green marbles. If three marbles are drawn out of the bag without replacement, what is the probability, to the nearest 10th of a percent, that all three marbles drawn will be red?

Answers

SOLUTION

Given the question, the following are the solution steps to answer the question.

STEP 1: Write the formula for probability

[tex]Probability=\frac{number\text{ of required outcomes}}{number\text{ of total possible outcomes}}[/tex]

STEP 2: Write the outcomes of the events

[tex]\begin{gathered} number\text{ of red marbles}\Rightarrow n(red)\Rightarrow8 \\ number\text{ of blue marbles}\Rightarrow n(blue)\Rightarrow7 \\ number\text{ of green marbles}\Rightarrow n(green)\Rightarrow6 \\ number\text{ of total marbles}\Rightarrow n(total)\Rightarrow21 \end{gathered}[/tex]

STEP 3: Write the formula for getting the probability that all three marbles drawn will be red

[tex]Pr(Red\text{ and Red and Red\rparen}\Rightarrow Pr(red)\times Pr(red)\times Pr(red)[/tex]

STEP 4: Calculate the probability

[tex]\begin{gathered} Pr(all\text{ three are reds\rparen}\Rightarrow\frac{8}{21}\times\frac{7}{20}\times\frac{6}{19} \\ =\frac{336}{7980}=0.042105263 \\ To\text{ percentage will be to multiply by 100} \\ 4.210526316\% \\ To\text{ the nearest tenth will be:} \\ \approx4.2\% \end{gathered}[/tex]

Hence, the probability, to the nearest 10th of a percent, that all three marbles drawn will be red is 4.2%

A data set is summarized in the frequency table below. Using the table, determine the number of values less than or equal to 6.ValueFrequency152332435463788397108113Give your answer as a single number. For example if you found the number of values was 14, you would enter 14.

Answers

The number of values less than or equal to 6 is 5 + 3 +2 +3 +4 +3 = 20

Sara has saved $500 and wants to buy a new computer. the computer she wants costs $1400. her current job pays $17.50 per hour after taxes . use an inequality to describe the situation , solve the inequality and write a sentence describing what the solution means to Sara

Answers

She has already 500

x hours woked

pund we got:

7.50 x >= 900

And inequality could be

fx:o

[tex]500+17.5x\ge1400[/tex]

because we need to gain 1400 or more

hours in order to have enough money to buy the computer

Sara needs to work at least 52 00/17.5

x >= 5r t o

The equatu

And then we can solve for x a

500+17.5x>>= 1400

utioen x

For this50 case we can do this:

500+ 17.

In the figure to the right, ABC and ADE are similar. Find the length of EC.

The length of EC is ___.

Answers

Answer:

ninety 90 feet or foot long

Rewrite y + 1 = -2x – 3 in standard form

Answers

The algebraic expressions can be written as

[tex]a+b+c=0[/tex]

The given expression is,

[tex]\begin{gathered} y+1=-2x-3 \\ -2x-y=1+3 \\ -2x-y=4 \\ -2x-y-4=0 \\ 2x+y+4=0 \end{gathered}[/tex]

Zales sells diamonds for $1,100 that cost $800. What is Zales’s percent markup on selling price? Check the selling price.

Answers

Zales's percent markup on the selling price as required in the task content is; 37.5%.

Percentages and markup price

It follows from the task content that the percent markup on the selling price be determined according to the given data.

Since the cost of diamonds is; $800 while the diamonds sell for $1,100. It follows that the markup on the selling price of the diamonds is;

Markup = Selling price - Cost price.

Hence, we have;

Markup = 1,100 - 800.

Therefore, the markup is; $300.

On this note, the percent markup can be determined as follows;

= (300/800) × 100%.

= 37.5%.

Ultimately, the percent markup on the diamonds is: 37.5%.

Answers

Explanation

Let the number be x. Therefore, we will have

[tex]\begin{gathered} 5x+7=8 \\ 5x=8-7 \\ 5x=1 \\ x=\frac{1}{5} \end{gathered}[/tex]

23 – 4u < 11 what is the answer

Answers

23 - 4u < 11

23 is adding on the left, then it will subtract on the right

-4u < 11 - 23

-4u < -12

-4 is multiplying on the left, then it will divide on the right. Remember that dividing by a negative number changes the sign.

u > (-12)/(-4)

u > 3

A straight line l1 with equation 5x - 7 = 0 cuts the x axis at point A. Straight line l2 is perpendicular to straight line l1 and passes through point A. What is the coordinates of point A and the equation of the straight line l2?

Answers

Point A has a coordinate of (7/5, 0) while the straight line l2 is represented by the equation y = 0

The coordinates of point A

The equation of line l1 is given as

5x - 7 = 0

It cuts the x-axis at point A

This means that

5A - 7 = 0

Solve for A

5A = 7

So, we have

A = 7/5

Rewrite as

A = (7/5, 0)

The equation of the straight line l2

From the question, we have

Lines l1 and l2 are perpendicular lines

The equation 5x - 7 = 0 has no y variable

So, the slope is undefined

The slopes of perpendicular lines are represented as follows

Slope 1 * Slope 2 = -1

So, we have

Slope 2 = -1/Slope 1

This gives

Slope 2 = -1/undefined

Evaluate

Slope 2 = 0

This means that l2 has a slope of 0

The equation of l2 is calculated as

y = m(x - x₁) + y₁

In this case,

A = x₁ and y₁ = 0

So, we have

y = m(x - A)

This gives

y = 0 * (x - 7/5)

Evaluate

y = 0

Hence, the equation of the straight line l2 is y = 0

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can you please give me any examples on how to do this

Answers

we can take two numbers of the sequence and subtract them to see the difference

[tex]1.9-1.2=0.7[/tex]

the sequence adds 0.7 each step

the next 3 terms are

[tex]3.3+0.7=4[/tex][tex]4+0.7=4.7[/tex][tex]4.7+0.7=5.4[/tex]

The diameter of circle is 20 inches. find the circumference in terms of pi

Answers

The below formula is used to find the circumference of a circle;

[tex]C=2\pi r[/tex]

But we know that the diameter of a circle is expressed as;

[tex]d=2r[/tex]

Let's replace 2r with d in the 1st equation, we'll then have;

[tex]C=\pi d[/tex]

We've been told that the diameter of the circle is 20inches, if we substitute this value into our equation, we'll have;

[tex]C=20\pi[/tex]

Solve the following system of equations by graphing3x+5y=10y=-x+4

Answers

ANSWER

The point of intersection of the two equations is (5, - 1)

The graph is

STEP BY STEP EXPLANATION

Step 1: The given equations are:

3x + 5y = 10

y= -x + 4

Step 2: Assume values for x in a table (example -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5) to determine the corresponding values for y for both equations

Step 3: Graph the equations and locate the intersection of the two equations

Describe a series of transformations that takes triangle ABC to triangle A’B’C’

Answers

Notice that if the triangle ABC is reflected over the X axis (red), and then reflected over the Y axis (green), we just would have to translate the triangle two units to the right (blue) to get A'B'C':

In basketball, " one on one" free throw shooting ( commonly called foul shooting) is done as follows: if the player makes the first shot(1point), she is given a second shot. If she misses the first shot, she is not given a second shot. Christine, a basketball player, has a 70% free throw record. (she makes 70% of her free throws). Find the probability that, given one-on-one free throw shooting opportunity, Christene will score one point.

Answers

If she will be able to shoot the first shot and miss the second shot, then she will obtain 1 point.

Thus, the probability that Christine will get the first shot is as follows:

[tex]P(1pt)=(0.7)(0.3)=0.21[/tex]

where the first factor is the probability that she will shoot the first shot and the second factor is the probability that she missed the second shot. Thus, the probability of obtaining 1 point is 21% or 0.21.

Absolute risk is defined as the proportion or percentage of people in a group for whom an undesirable event occurs. In college classrooms, students typically can choose their own seats. Professors have noticed a difference in grades between students who choose to sit in the front and those who choose to sit in the back. For example, in one math class, 9 of the 20 students who sat in the back failed the class, but only 3 of the 20 students who sat in the front failed the class. What was the absolute risk of failing the class for students who sat in the back? For students who sat in the front? Give your answers as fractions, proportions, and percents.

Answers

Given in the scenario:

a.) 9 of the 20 students who sat in the back failed the class.

b.) 3 of the 20 students who sat in the front failed the class.

A.) The absolute risk of failing the class for students who sat in the back.

In the back, 9 of the 20 students who sat in the back failed the class.

The absolute risk in proportion = 9:20

The absolute risk in fraction = 9/20

The absolute risk in percentage = (9 ÷ 20) x 100 = 0.45 x 100 = 45%

B.) The absolute risk of failing the class for students who sat in the front.

In the front, 3 of the 20 students who sat in the front failed the class.

The absolute risk in proportion = 3:20

The absolute risk in fraction = 3/20

The absolute risk in percentage = (3 ÷ 20) x 100 = 0.15 x 100 = 15%

Which expression is undefined? O A. 11 B.- 3 C.6-6) D. -4+0

Answers

Answer:

Option C

Step-by-step explanation:

Undefined expression:

Division by 0, or fraction in which the denominator is 0. In this question, this is in option C, since 3/(6-6) = 3/0.

3(4x+1)^2-5=25 using square root property

Answers

Answer:

[tex]x=\frac{-1+\sqrt{10}}{4}\text{ or }x=\frac{-1-\sqrt{10}}{4}[/tex]

Explanation:

Given the equation:

[tex]3\left(4x+1\right)^2-5=25[/tex]

To solve an equation using the square root property, begin by isolating the term that contains the square.

[tex]\begin{gathered} 3(4x+1)^{2}-5=25 \\ \text{ Add 5 to both sides of the equation} \\ 3(4x+1)^2-5+5=25+5 \\ 3(4x+1)^2=30 \\ \text{ Divide both sides by 3} \\ \frac{3(4x+1)^2}{3}=\frac{30}{3} \\ (4x+1)^2=10 \end{gathered}[/tex]

After isolating the variable that contains the square, take the square root of both sides and solve for the variable.

[tex]\begin{gathered} \sqrt{(4x+1)^2}=\pm\sqrt{10} \\ 4x+1=\pm\sqrt{10} \\ \text{ Subtract 1 from both sides} \\ 4x=-1\pm\sqrt{10} \\ \text{ Divide both sides by 4} \\ \frac{4x}{4}=\frac{-1\pm\sqrt{10}}{4} \\ x=\frac{-1\pm\sqrt{10}}{4} \end{gathered}[/tex]

Therefore, the solutions to the equation are:

[tex]x=\frac{-1+\sqrt{10}}{4}\text{ or }x=\frac{-1-\sqrt{10}}{4}[/tex]

which methods correctly solve for the variable x in the equation 2/5m = 8?

Answers

Ok, so the equation is (2/5)m=8

1st option: Divide by 2 on both sides, then multiply by 5 on both sides:

[tex]\begin{gathered} \frac{2}{10}m=4 \\ \frac{10}{10}m=20 \\ m=20 \end{gathered}[/tex]

2nd option: Multiply both sides by 5/2

[tex]\begin{gathered} \frac{2}{5}\cdot\frac{5}{2}m=8\cdot\frac{5}{2} \\ m=20 \end{gathered}[/tex]

3rd option: First dristibute 2/5 to (m=8), the multiply by 5/2 in both sides

[tex]\begin{gathered} \frac{2}{5}m=8 \\ \frac{5}{2}\cdot\frac{2}{5}m=8\cdot\frac{5}{2} \\ m=20 \end{gathered}[/tex]

4th option: Divide both sides by 2/5:

[tex]\begin{gathered} \frac{\frac{2}{5}}{\frac{2}{5}}m=8\cdot\frac{5}{2} \\ m=20 \end{gathered}[/tex]

5th option: First, multiply by 5. Then, divide by 2.

[tex]\begin{gathered} 5\cdot\frac{2}{5}m=40 \\ 2m/2=40/2 \\ m=20 \\ \end{gathered}[/tex]

All the methods are correct

Which recipe makes more cookies per cup of chocolate chips?

Answers

We are given the recipes of Grandma and Betty Potter box.

We are asked to find out which of them makes more cookies per cup of chocolate chips.

Let us find the unit rate for both of them and compare which is greater.

Grandma:

She uses 1 1/2 cups of chocolate chips to make 24 cookies.

The unit rate is

[tex]\frac{24}{1\frac{1}{2}}=\frac{24}{\frac{3}{2}}=24\times\frac{2}{3}=16[/tex]

So the unit rate is 16 cookies per chocolate chip.

Betty Potter box:

60 cookies are made using 3 chocolate chips.

80 cookies are made using 4 chocolate chips.

200 cookies are made using 10 chocolate chips.

The unit rate is

[tex]\begin{gathered} \frac{60}{3}=20 \\ \frac{80}{4}=20 \\ \frac{200}{10}=20 \end{gathered}[/tex]

So the unit rate is 20 cookies per chocolate chip.

As you can see, the recipe of Better Potter box makes more cookies per cup of chocolate chip.

Determine whether the ratios are equivalent.
2:3 and 24:36
O Not equivalent O Not equivalent

Answers

We can conclude that the given ratios 2:3 and 24:36 are equivalent.

What are ratios?A ratio in math displays how many times one number is contained in another. The ratio of oranges to lemons, for instance, is eight to six if there are eight oranges and six lemons in a bowl of fruit. The proportions of oranges to the total amount of fruit are 8:14 for oranges and 6:8 for lemons, respectively.

So, ratios are equivalent or not:

2:3 and 24:362/3 = 24/362/3 = 2/3 (Divide by 12)

Then, 2:3 :: 2:3

Therefore, we can conclude that the given ratios 2:3 and 24:36 are equivalent.

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octavius wants to write the equation of a line perpendicular to y=-4x + 5 that passes through the point (8,-3). Describe the mistake octavius made and write the correct equation of the line.

Answers

The equation of line perpendicular to 4y = x-8 passing through (4,-1) is:

[tex]y = \frac{1}{4} x-5[/tex].

What is a equation of line?

These lines are written in the form y = mx + b, where m is the slope and b is the y-intercept. We know from the question that our slope is 3 and our y-intercept is –5, so plugging these values in we get the equation of our line to be y = 3x – 5.

Given equation of line is:

y=-4x + 5

Let [tex]m_{1}[/tex] be the slope of given line

Then,

[tex]m_{1}[/tex] = -4

Let [tex]m_{2}[/tex] be the slope of line perpendicular to given line

As we know that product of slopes of two perpendicular lines is -1.

[tex]m_{1}*m_{2} = -1\\- 4*m_{2}=-1\\ m_{2} = \frac{1}{4}[/tex]

The slope intercept form of line is given by

[tex]y = m_{2}x+c[/tex]

[tex]y = \frac{1}{4} x+c[/tex]

to find the value of c, putting (4,-1) in equation

[tex]-3= \frac{1}{4} *8+c\\-3-2 = c\\c = -5[/tex]

Putting the value of c in the equation

[tex]y=\frac{1}{4} x-5[/tex]

Hence, The equation of line perpendicular to 4y = x-8 passing through (4,-1) is [tex]y=\frac{1}{4} x-5[/tex].

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xyx2xy1645256720 2258 484 1,2762873 7842,044 3294 1,0243,008 45141 2,025 6,345 ∑x=143 ∑y=411 ∑x2=4,573 ∑xy=13,393 Which regression equation correctly models the data?y = 2.87x + 0.12y = 2.87x + 11.85y = 3.39x – 14.75y = 3.39x – 9.24

Answers

We are asked to identify the correct regression equation.

The regression equation is given by

[tex]y=bx+a[/tex]

Where the coefficients a and b are

[tex]a=\frac{\sum y\cdot\sum x^2-\sum x\cdot\sum xy}{n\cdot\sum x^2-(\sum x)^2}[/tex][tex]b=\frac{n\cdot\sum xy-\sum x\cdot\sum y}{n\cdot\sum x^2-(\sum x)^2}[/tex]

Where n is the number of observations that is 5.

Let us substitute the following into the above formula.

∑x=143

∑y=411

∑x^2=4,573

∑xy=13,393

[tex]a=\frac{411\cdot4573-143\cdot13393}{5\cdot4573-(143)^2}=-14.75[/tex][tex]b=\frac{5\cdot13393-143\cdot411}{5\cdot4573-(143)^2}=3.39[/tex]

So, the coefficients are

a = -14.75

b = 3.39

Therefore, the correct regression equation is

[tex]y=3.39x-14.75[/tex]

Describe the association in the scatter plot below.----------------The scatter plot shows (positive linear, positive linear with one outlier, negative linear, negative linear with one outlier, nonlinear, or no) association because as the plotted values of x increase, the values of y generally (decrease, increase, show no pattern or follow a nonlinear pattern).

Answers

From the given figure

The given point can form a line with a negative slope, because when

the values of x increase the values of y decrease

Then the scatter plot shows a negative linear association because

as the values of x increase, the values of y generally decrease

A statement of Chandler's biweekly earnings is given below. What is Chandler's gross pay?

Answers

SOLUTION:

Step 1:

In this question, we are asked to calculate Chandler's gross pay from the statement of bi-weekly earnings.

Step 2:

To get the Gross pay, we need to do the following:

[tex]\text{Gross pay - Total Deductions = Net Pay}[/tex]

Now, we need to calculate Total Deductions:

[tex]\text{ \$ 105.00 + \$ 52.14 + \$ 10.62 + \$ 26. 15 = \$ 193.91}[/tex]

Now, we have that the Net Pay = $ 780. 63

Then,

[tex]\begin{gathered} \text{Gross Pay - \$ 193. 91 = \$ 7}80.\text{ 63} \\ \text{Gross pay = \$ 780.63 + \$ 193.91} \\ \text{Gross Pay = \$ 974. 54} \end{gathered}[/tex]

CONCLUSION:

Chandler's Gross Pay = $ 974. 54

consider the parent function f(x)=x^2. a. graph y=f(x). b. write an equation for f(1/2x). Then sketch a graph of y=f(1/2x) and describe the transformation. c.write an equation for f(3x). Then sketch a graph of y=f(3x) and describe the transformation.

Answers

In this case, we'll have to carry out several steps to find the solution.

Step 01:

Data

f(x)=x²

f(1/2x) = ?

graph of y=f(1/2x) = ?

Step 02:

b. f(1/2x)

x ===> 1/2x

[tex]f\text{ (1/2 x) = (}\frac{1}{2}x)^2=\frac{1}{4}x^2[/tex]

Step 03:

c. Graph:

We give values to x, and we obtain the values of y.

f(x) = 1/4 x²

e.g.

if x = 4

y = 1/4 (4)² = 1/4 * 16 = 4

That is the solution for b. and c.

let f ( x ) = 6356 x + 5095 . Use interval notation. Many answers are possible.

Answers

The equation of the function has its domain representation in interval notation as (oo, oo)

How to determine the domain of the function

From the question, the equation of the function is given as

f ( x ) = 6356 x + 5095

Rewrite the equation of the function properly by removing the excess spaces

So, we have

f(x) = 6356x + 5095

The above equation is a linear equation

A linear equation is represented as

f(x) = mx + c

As a general rule;

The domain of a linear equation is all set of real numbers

This is the same for the range

i.e. the range of a linear equation is all set of real numbers

When the set of real numbers is represented as an interval notation, we have the following representation

(oo, oo)

Hence, the domain is (oo, oo)