If the volume of the cylinder is 942 in³, what is the radius of the base of the cylinder? Use 3.14 for pie and enter your answer as a whole number.

Therefore, the height of the rectangle and the radius of the semicircle should be 8 feet each in order for the Norman window to admit the most light.

The distance along the line defining a half circle's boundary is known as the semi circle's perimeter. It is made up of the diameter of the semi circle, which is a straight line that passes through the center of the circle, and the semi circle's curving arc. For example, he radius of the semicircle and the height of the rectangle such that the window will admit the most light. A semi circle's diameter must be determined before calculating the circumference. This may be accomplished by determining the radius of the semicircle, which is the separation between any two points on the circle and the circle's center. Simply said, the diameter is twice the radius.

How to solve?

If the perimeter of the Norman window is 32 feet, then each side of the rectangle must be 8 feet long. ( using 2(l+b))

The radius of the semicircle must also be 8 feet, as the perimeter of a semicircle is equal to 2 * pi * radius.

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By using the differentials to approximate the possible propagated error in computing the area of the end of the log is ±2π

The measurement of the radius of the end of a log = 8 inches

The possible error of radius = 1/8 inches

The area of cross section of the end of the log = πr^2

A =  πr^2

dA / dr = 2πr

dA = 2πr(dr)

Here the possible error of radius = 1/8 inches

dr = 1/8 inches

Substitute the values in the equation

dA = 2π × 8 × (± 1/8)

= ±2π

Therefore, the possible propagated error in computing the area is ±2π

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Number of children adults and students who attended the theatre is 182 , 91 , 94 respectively

Let Number of Children in theatre = c

Let Number of Adults in theatre = a

Let Number of Students in Theatre =s

Total money earned by selling the tickets = $2660

According to the given information

c + s + a = 367

a = c/2

5c + 7s + 12a = 2660

5c + 5s + 5a = 5(367)= 1835

Using elimination and substitution method

subtract

2s +7a = 2660-1835 = 825

2s +7(c/2) = 2s +7c/2 = 825

4s +7c = 1650

c + s + c/2 = 367

2c + 2s + c =734

2s + 3 c =734

4s + 6c=1468

4s + 7c =1650

c = 182 children

a= 91 adults

s = 367-273 = 94 students

Number of children adults and students who attended the theatre is 182 , 91 , 94 respectively

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Answer:

$150 per month

Step-by-step explanation:

Susan plans to attend college in 3 years. She will need $ 17,400 for the first-year cost. Her uncle gave her a special gift of $12,000 to use toward the cost. How much must Susan save each month to have enough for the first-year cost?

17,400 - 12,000 = $5,400 Susan needs to pay in 3 years.

3 years = 36 months

$5,400/36 = $150 per month

The amount of interest he will pay on this loan$12,400.

What is  interest?

The fee you pay to borrow money or the fee you charge to lend money is called interest. The most common way to represent interest is as an annual percentage of the loan amount. The interest rate on the loan is known as this percentage.

Principal = $10,000

Rate = 8%

Time = 36 months, 1 year - 12 months

36 months - 36/12= 3 years

S.I. = PRT/100

10,000*8*3/100

240,000/100 = $2,400

A = P+I = 10,000+2,400= $12,400

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The solution of the system of equation y = -x + 3 and y = 3x + 7 will be (-1,4).

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of 2 mathematical expressions.

More than one variable may be present inside a linear equation. An equation is said to be linear if the maximum power of the variable is consistently unity.

As per the given system of equations,

y = -x + 3 and y = 3x + 7

Compare both,

3x + 7 = -x + 3

3x + x = 3 - 7

x = -1

y = -(-1) + 3 = 4

So the solution will be (-1,4).

Hence "The system of equations y = -x + 3 and y = 3x + 7 will have the following solution: (-1,4).".

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The time when the money will double is 11.6 years

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

When the money doubles, the amount is

A = 900 * 2

A = 1800

The compound interest formula is given as

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

Also from the question, we have

n = 12

This is so because the loan is compounded 12 times in a year as a result of being compounded monthly

Solving further, we replace the variables with their values in the above equation

1800 = 900 * (1 + (6%)/12)^(t*12)

Divide both sides by 900

2 = (1 + (6%)/12)^(t*12)

Also, we have

2 = (1.005)^(t*12)

Take the logarithm of both sides

12t = log(2)/log(1.005)

Evaluate the quotients

12t = 139

Divide both sides by 12

t = 11.6

Hence, the number of years is 11.6 years

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The difference between the boards’ total lengths is  86 inches.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

The difference between the total lengths of the boards is the positive value when we subtract one from another.

The total length of the red cedar board is = (8×10.25) = 82 inches.

The total length of the tiger maple board is = (16×10.5) = 168 inches.

∴ The difference is (168 - 82) = 86 inches.

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Hello,

I hope you and your family are doing well!

The function y = |20x+15| is an absolute value function. Absolute value functions have the form y = |f(x)|, where f(x) is a linear function.

In this case, the linear function is 20x+15, so the absolute value function is:

y = |20x+15|

The absolute value function takes the absolute value of the output of the linear function. This means that if the output of the linear function is positive, the absolute value function returns the same value. If the output of the linear function is negative, the absolute value function returns the opposite of that value.

For example, if x = -1, then the output of the linear function is 20(-1)+15 = -5. The absolute value of -5 is 5, so the absolute value function returns y = 5.

On the other hand, if x = 1, then the output of the linear function is 20(1)+15 = 35. The absolute value of 35 is 35, so the absolute value function returns y = 35.

Overall, the absolute value function returns the distance of the output of the linear function from 0, without considering the sign of the output.

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Happy Holidays!

The information provided indicates that the means test is too narrow because it excludes factors like the type of engineer, amount of education, and experience. The type of engineer or educational level may be a reflection of the gender with lower earnings.

Additional information on the factors, including gender, education, and the type of engineer, could enhance the research.

Then, it is advised to build a multiple regression where the wage is the dependent variable and the other four variables are independent variables. The "difference in means" test is not appropriate for identifying gender bias in salary setting due to the significance of the omitted variable.

The variables which  are likely useful to add to the regression to control for important omitted​ variables are excessive drug or alcohol use and their gang activity.

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Answer: Muffins > 15 (Greater than 15)

Step-by-step explanation: If the most Scott can eat without feeling sick is 15 muffins, this means that any more muffins would make him sick. The amount of muffins to make Scott sick would have to be greater than 15.

The intersection of the curve with the x-axis will be at (−1, 0) and (4, 0) for each factor.

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

The function written in factored form is given below.

f(x) = (−x + 4) (2x + 2)

For the factor (−x + 4), we have

−x + 4 = 0

x = 4

For the factor (2x + 2), we have

2x + 2 = 0

2x = −2

x = −1

The intersection of the curve with the x-axis will be at (−1, 0) and (4, 0) for each factor.

The graph is given below.

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The graph of the linear equation y = (3/2)*x - 7

Can be seen in the image at the end.

A general linear equation is written as:

y = a*x + b

Where a is the slope and b is the y-intercept.

Here we know that the slope is -2 and the y-intercept is -4, then the linear equation is:

y = -2x - 4

To graph that line we need to find two points on it, to get the points we need to replace the value of x.

if x = 0

y = -2x - 4

y = -4

So we have the point (0, -4)

And if x = 1:

y = -2*1 - 4

y = -6

So we have the point (1, -6)

Now we just need to graph these two points and connect them with a line.

You can see the graph below.

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Answer: A. 6m

Step-by-step explanation: the graph shows at the x axis that it is time. so, go to two seconds and then go up to where the line is. it goes up to 6m so that is the answer.

Answer:

x = 10.47

Step-by-step explanation:

circumference = 2 * 10 * π = 20π

centre angle of a circle (in radiant) = 2π

x : 20π = π/3 : 2π

x = (20π * π/3)/ 2π

x = 10 * π/3

x = 10π/3

x = 10.466667

x = 10.47

The equations are 3x - 3y = 4 and 3x + 2y = -6.

What is a system of linear equations in matrices?

A system of linear equations can be represented in matrix form using a coefficient matrix, a variable matrix, and a constant matrix. Consider the system, 2x+3y=85 and x−y=−2. The coefficient matrix can be formed by aligning the coefficients of the variables of each equation in a row.

In the given matrix,

The first equation would be

3x - 3y = 4

and the second equation would be

3x + 2y = -6

Hence, the equations are 3x - 3y = 4 and 3x + 2y = -6.

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Step-1) Determine the value of the x and y variables

To verify if (2, -3) is a solution to the system of equations provided in the question, we will need to determine the value of the x and y variables.

To do this, we will have to compare the coordinates of the point (2, -3) with the "original form" of the coordinate point (x, y). Then, we will have to identify the values (from the coordinates) in their respective order.

Example: (4, 5) = (x, y) ⇒ x = 4; y = 5

Step-2) Substitute the x and y values into one of the equations

The next step is to substitute the value of the variables. This is a required step for the coordinate provided to be verified. Once you substitute the value of the variables, you can move on to the next step process.

Step-3) Simplify the equation completely until it is in simplified form

Then, we will have to simplify the equation completely.

This is also a required step because simplifying is a process where we can convert an expression to a specific numerical value, which can help us create final conclusions to a problem. It can also help us obtain a specific numerical value from any large/small expression provided.

Step-4) Compare both sides and verify if they are equal/not equal

If both sides of the equation do not receive the same value, then the point (2, -3) is not a solution to the provided system of equation.

Given Information (Provided in the question):

Step-1) Determine the x and y variables from the coordinates:

Step-2) Substitute the variables into any equation:

Step-3) Simplify both sides of the equation:

Therefore, (2, -3) is not a solution to the system of equations.

Answer:

I think 50

Step-by-step explanation:

The value of z in the equation 1/3(6z + 7.2) = 0.5(8z + 2) - 0.4 is z=0.9

How to solve the equation?

1/3(6z + 7.2) = 0.5(8z +2) -0.4

(6/3)z + 7.2/3 = (8/2)z + 2/2 - 0.4

2z + 2.4 = 4z + 1 - 0.4

2z - 4z = 0.6 - 2.4

-2z = -1.8

z= 0.9

Hence, the value of z in the equation 1/3(6z + 7.2) = 0.5(8z + 2) - 0.4 is z=0.9

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Answer:

1,2,3,5,6,10,15,30

Step-by-step explanation:

1*30

2*15

3*10

5*6

The meaning of the given formula is that the first term (a₁) is equal to the nth term (aₙ) of a sequence multiplied by 10 and it is also equal to the preceding nth term (aₙ₋₁) of a sequence plus four (4).

In Mathematics, an arithmetic sequence can be defined as a series of real and natural numbers in which each term differs from the preceding term by a constant numerical quantity.

Mathematically, the nth term of an arithmetic sequence can be calculated by using this expression:

aₙ = a₁ + (n - 1)d

Where:

Based on the information provided, we can logically deduce that the first term (a₁) of the arithmetic sequence can either be calculated by multiplying the nth term (aₙ) by ten 10 or adding four (4) to the preceding nth term (aₙ₋₁) of the arithmetic sequence.

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Answer: proved

Step-by-step explanation:

Prove the inequality:

[tex]\displaystyle\\\frac{(a+b)^2}{c} +\frac{c^2}{a} \geq 4b\ \ \ \ \ a > 0\ \ \ \ \ b > 0\ \ \ \ \ c > 0\\[/tex]

Multiply both parts of the inequality by ac (a>0, c>0):

[tex]a(a+b)^2+c*c^2\geq 4abc[/tex]

Simplify the left side of the inequality using the Cauchy inequality:

                                  [tex]\boxed {t+v\geq 2\sqrt{tv} }[/tex]      

[tex]a(a+b)^2+c(c^2)\geq 2\sqrt{a(a+b)^2c(c^2)} \\\\a(a+b)^2+c(c^2)\geq 2(a+b)c\sqrt{ac} \\\\a(a+b)^2+c(c^2)\geq 2(ac+bc)\sqrt{ac}\ \ \ \ \ (1)[/tex]

Simplify the right side of the inequality using the Cauchy inequality:

[tex]ac+bc\geq 2\sqrt{acbc} \\\\ac+bc\geq 2\sqrt{abc^2} \\\\ac+bc\geq 2c\sqrt{ab}[/tex]\ \ \ \  \ (2)

Substitute expression (2) into expression (1):

[tex]a(a+b)^2+c(c^2)\geq 2*2c\sqrt{ab} \sqrt{ab} \\\\a(a+b)^2+c(c^2)\geq4abc[/tex]

Hence,

[tex]\displaystyle\\\frac{(a+b)^2}{c} +\frac{c^2}{a} \geq 4b[/tex]

The height of a nearby tree is 15ft

What is Application of trigonometry ?

Trigonometry is the study of lengths, heights, and angles through computations involving triangles. There are many applications for trigonometry and its functions in daily life. For instance, it is employed in astronomy to gauge the distances between nearby stars, in geography to gauge the separations between landmarks, and in the satellite navigation system.

According to question

let

The height of tree = h

Angle elevation = α

Given

Length of shadow of tree = 20 ft

height of man = 6 ft

Length of shadow of man = 8 ft

We know that

tanθ = Perpendicular/Base

therefore:

tanα = 6/8 = h/20

⇒ 6*20 = 8h

⇒ 120 = 8h

⇒ h = 120/8

⇒ h = 15

hence, the height of a nearby tree is 15 ft.

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By knowing that the cost equation must be a proportional relation, we will find that the equation is:

y = $0.64*x

The cost equation is proportional, if we define y as the cost and x as the number of pounds, the equation is something like:

y = k*x

We know that for (5 + 1/2) pounds the cost is $3.52, then we can replace these values in the equation above to get.

$3.52 = k*(5 + 1/2)

$3.52/(5 + 1/2) = k = $0.64

Then the cost equation for x pounds of potatoes is:

y = $0.64*x

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The solution for the given system of inequalities is (5,-2). which is the correct answer would be an option (A).

The first inequality, 2x + y > 3, can be rewritten in slope-intercept form as y > -2x + 3. This represents a line with a slope of -2 and a y-intercept of (0,3).

The second inequality, x - y > 1, can also be rewritten in slope-intercept form as y < x - 1. This represents a line with a slope of 1 and a y-intercept of (0,-1).

When you graph these two lines, you will get two lines that intersect at some point. The region of the graph that satisfies both inequalities will be the area that is bounded by these lines and lies above or below them, depending on the inequality symbol.

In this case, the solution to the system of inequalities is the region of the graph that is above the line y = -2x + 3 and below the line, y = x - 1. This region is shaded in the graph below.

From the graph, you can see that the solution includes all points in the shaded region. You can also see that points B (0,-5), C (0,0) , and D (0,5) is not included in the solution, while point A (5,-2) included in the solution.

Therefore, the correct answer is A (5,-2).

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Answer:

425.39 ft

Step-by-step explanation:

[tex]\frac{1}{2}mv^2=mgh\\\frac{1}{2}v^2=gh\\v^2=2gh[/tex]

h = [tex]\frac{v^2}{2g} = \frac{165^2}{2 * 32}=\frac{27225}{64}=425.39ft[/tex]

After 11.71 years the amount will double to the principal amount  

Compound interest refers to the interest added to a loan or deposit that depends on both the principal amount and compound interest rate and the number of compoundings in a year.

The formula for the amount in compound interest is given by

where P = Principal amount

r = Rate of interest

n = Number of compounds

t = Time period  

From the given data

Principal amount, p = $ 2500

Rate of interest, r = 6% = 6/100 = 0.06

Number of compounds, n = 2    [ ∵ 12/6 = 2 ]

Here we need to find the time period

Let's assume that after 't' years the amount will be doubled i.e $ 5000

From the given problem,

Amount, A = 2500(1+0.06/2)^2t        

= 2500(1+0.06/2)^2t  

= 2500(1 + 0.03)^2t  

= 2500 (1.03)^2t    

As we assumed after 't' years the amount is $ 5000

=>  2500 (1.03)^2t = 5000

=> (1.03)^2t = 5000/2500

=> (1.03)^2t = 2

Apply to log on both sides

=> log (1.3)^t  = log 2

=> 2t log (1.3) = log 2

=> 2t = log 2/ log(1.03)

=> 2t = 23.4497722

[ use calculator for log (1.03) and log 2 values]

=> t = 11.7248861  

=> t = 11.71 ≅ 11 years

Therefore,

After 11.71 years the amount will double to the principal amount

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After solving the provided question, Aisha spent 7.9 hours and 9.9 hours, respectively, working last week as a babysitter and a landscaper.

A mathematical statement that has at least two terms, both of which contain variables or equal integers, is defined as an equation.

Aisha is reportedly working two summer jobs, earning $7 an hour for babysitting and $8 an hour for gardening.

Let's say you were to watch the children for x hours.

And y would be the amount of hours spent on landscaping.

Last week, Aisha spent two more hours babysitting than a landscaper, earning a total of $134.

[tex]y = x + 2\\7x + 8y = 134\\Substitute the value of y = x + 2 in the equation,\\7x + 8(x + 2) = 134\\7x + 8x + 16 = 134\\15x = 134 - 16\\15x = 118\\x = 118/15\\x = 7.9\\Substitute the value of x = 7.9 in the equation y = x + 2 ,\\y = 7.9 + 2\\y = 9.9\\[/tex]

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