Equations of circles - HiSet: High School Equivalency Test: Math

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Question

Find the area of a square with the following side length:

Answer

We can find the area of a circle using the following formula:

In this equation the variable, , represents the length of a single side.

Substitute and solve.

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Question

The perimeter of a square is . In terms of , give the area of the square.

Answer

Since a square comprises four segments of the same length, the length of one side is equal to one fourth of the perimeter of the square, which is $\frac{1}{4}P$ . The area of the square is equal to the square of this sidelength, or

$\left (\frac{1}{4}P \right )^{2} = \left (\frac{1}{4} \right )^{2} P^{2} = \frac{1}{16}P^{2}$ .

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Question

The volume of a sphere is equal to $108 \pi$ . Give the surface area of the sphere.

Answer

The volume of a sphere can be calculated using the formula

$V= \frac{4}{3} \pi r^{3}$

Solving for :

Set $V = 108 \pi$ . Multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 108 \pi$

$\pi r^{3} = 81\pi$

Divide by $\pi$ :

$\frac{\pi r^{3} }{\pi}= \frac{81\pi}{\pi}$

$r^{3} = 81$

Take the cube root of both sides:

$r = \sqrt[3]{81} = \sqrt[3]{27(3)} = \sqrt[3]{27} \cdot \sqrt[3]{3} = 3 \sqrt[3]{3}$

Now substitute for in the surface area formula:

$A = 4 \pi r^{2}$

$A = 4 \pi (3 \sqrt[3]{3})^{2}= 4 \pi (3 )^{2}( \sqrt[3]{3})^{2} = 4 \pi (9) ( \sqrt[3]{3^{2}}) = 36 \pi \sqrt[3]{9}$ ,

the correct response.

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Question

Express the area of a square plot of land 60 feet in sidelength in square yards.

Answer

One yard is equal to three feet, so convert 60 feet to yards by dividing by conversion factor 3:

$60 \textup{ ft} \div 3 \textup{ ft / yd}= 20 \textup{ yd}$

Square this sidelength to get the area of the plot:

$20 \textup{ yd} \times 20 \textup{ yd} = 400 \textup{ yd} ^{2}$ ,

the correct response.

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Question

A square has perimeter . Give its area in terms of .

Answer

Divide the perimeter to get the length of one side of the square.

$s= \frac{P}{4}$

$s= \frac{12y+4}{4}$

Divide each term by 4:

$s= \frac{12y}{4}+ \frac{4}{4} = 3y+ 1$

Square this sidelength to get the area of the square. The binomial can be squared by using the square of a binomial pattern:

$A= s^{2}$

$=( 3y+ 1)^{2}$

$=( 3y )^{2}+ 2 (3y)(1)+ 1^{2}$

$= 3^{2}y ^{2}+ 2 (3 )(1)y+ 1^{2}$

$=9y ^{2}+ 6y+ 1$

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Question

A cube has surface area 6. Give the surface area of the sphere that is inscribed inside it.

Answer

A cube with surface area 6 has six faces,each with area 1. As a result, each edge of the cube has length the square root of this, which is 1.

This is the diameter of the sphere inscribed in the cube, so the radius of the sphere is half this, or $r= \frac{1}{2}$ . Substitute this for in the formula for the surface area of a sphere:

$A = 4\pi r^{2} = 4\pi \left ( \frac{1}{2} \right )^{2} = 4\pi \left ( \frac{1}{4} \right ) = \pi$ ,

the correct choice.

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Question

Find the area of a circle with the following radius:

Answer

The area of a circle is found using the following formula:

$A=\pi r^2$

In this formula the variable, , is the radius. Let's substitute in our known values and solve for the area.

$A=\pi 6^2$

$A=36\pi$

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Question

Give the area of the circle on the coordinate plane with equation

$x^{2}+y^{2}+ 6x-20y+ 9=0$ .

Answer

We must first rewrite the equation of the circle in standard form

$(x-h)^{2} + (y-k)^{2}= r^{2}$ ;

will be the radius.

$x^{2}+y^{2}+ 6x-20y+ 9=0$

Subtract 9 from both sides, and rearrange the terms remaining on the left as follows:

$x^{2}+y^{2}+ 6x-20y+ 9- 9 =0- 9$

$x^{2}+ 6x + \underline{; ; ; ; ; ; }+y^{2}-20y+ \underline{; ; ; ; ; ; } = - 9$

Note that blanks have been inserted after the linear terms. In these blanks, complete two perfect square trinomials by dividing linear coefficients by 2 and squaring:

$(6 \div 2)^{2} = 3^{2} = 9$

$(20 \div 2)^{2} = 10^{2} = 100$

Add to both sides, as follows:

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = - 9+9+ 100$

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = 100$

Rewrite the expression on the left as the sum of the squares of two binomials.

$(x+3)^{2} + (y-10)^{2} = 100$

The equation is now in standard form.

$r^{2} = 100$

The area of this circle can be found using this formula:

$A = \pi r^{2} = \pi (100) = 100 \pi$ ,

the correct response.

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Question

A circle on the coordinate plane has center and passes through . Give its area.

Answer

The radius of the circle is the distance between its center and a point that it passes through. This radius can be calculated using the distance formula:

$r = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$ .

Setting $x_{1} = 2, x_{2} = 10, y_{1}=9, y_{2} =3$ :

$r = \sqrt{(10-2)^{2}+(3-9)^{2} }$

$= \sqrt{8^{2}+(-6)^{2} }$

$= \sqrt{64+36}$

$= \sqrt{100}$

The area of a circle can be calculated by substituting 10 for in the equation

$A = \pi r^{2}$

$= \pi (10)^{2}$

$= 100 \pi$ ,

the correct response.

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Question

A farmer has to make a square pen to hold chickens. If each chicken has to have $1 ;\textup{m}^2$ of area to roam and there are chickens total, what is the length of the amount of fencing required to pen in the chickens?

Answer

A square has area formula

The total area required for the chickens will be

$25 ;\textup{m}^2$

since each chicken requires $1;\textup{m}^2$ of space and there are chickens.

Thus, we have

for the length of our chicken fence.

$\sqrt{25}=\sqrt{l^2}^$

Since there are four sides of a square and each side has a length of 5, the total length of fence required is

$5\times4 = 20;\textup{m}$ .

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Question

The average population density of a geographic area is defined to be the average number of residents per square mile.

Above is the map of a county whose population is about 120,000. Which of the following is the best estimate of the average population density?

Answer

The county is in the shape of a trapezoid with bases of length and , and with height . Its area in square miles can be found by substituting in the formula for the area of a trapezoid:

$A = \frac{1}{2} (B+b)h$

$A = \frac{1}{2} (44+33) (30)$

$A = \frac{1}{2} (77) (30)$

square miles

Divide the population by this area to obtain an estimate of the population density:

$120,000\div 1,155 \approx 44.8 \approx 103.9$ persons per square mile.

Of the given choices, 100 persons per square mile comes closest.

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Question

Source: United States Census Bureau

The average population density of a geographic area is defined to be the average number of residents per square mile.

Above is a table with the land areas and populations of five states.Which state among the five has the greatest population density?

Answer

For each state, divide the population by the land area. We can round each figure to the nearest whole for simplicity's sake.

Alabama:

$4,860,000 \div 50,645 \approx 96$ persons per square mile.

Arkansas:

$2,980,000 \div 52,035 \approx 57$ persons per square mile.

Kentucky:

$4,430,000 \div 39,486 \approx 112$ persons per square mile.

Mississippi:

$2,990,000 \div 46,923 \approx 64$ persons per square mile.

Tennessee:

$6,600,000 \div 41,235 \approx 160$ persons per square mile.

Tennessee has the greatest population density among the five states.

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Question

Source: United States Census Bureau

The average population density of a geographic area is defined to be the average number of residents per square mile.

Above is a table with the population densities and the land areas of five states. Of the five, which state has the greatest population?

Answer

Multiply the population density of each state by its corresponding area to get an estimate of the population (round to the nearest thousand for simplicity's sake):

Alaska:

$1.3 \times 570,641 \approx 742,000$

North Dakota:

$11.0 \times 69,001 \approx 759,000$

South Dakota:

$11.1 \times 75,811 \approx 842,000$

Vermont:

$67.9 \times 9,217 \approx 626,000$

Wyoming:

$6.0 \times 97,093 \approx 583,000$

Of the five states, South Dakota is the most populous.

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Question

Find the circumference of a circle with the following diameter:

Answer

A circle's circumference is calculated using the following formula:

$C=2\pi r$

In this equation, the variable, , is the circle's radius.

In our problem we are given the diameter. The diameter is related to the radius in the following manner:

Let's rewrite our formula by substituting the diameter for the radius.

$C=d\pi$

Let's substitute and solve.

$C=14\pi$

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Question

Which of the following is closest to the length of a 270-degree arc of a circle with radius 20?

Answer

A circle with radius 20 has as its circumference $2 \pi$ times this, or

$C = 2 \pi r = 2 \pi (20 )= 40 \pi$

There are 360 degrees in a circle, so a 270-degree arc is

$\frac{270}{360} = \frac{270 \div 90}{360 \div 90} = \frac{3}{4}$

of the circle. Therefore, the length of this arc is

$L= \frac{3}{4} C = \frac{3}{4} (40 \pi)= 30 \pi$

$\pi \approx 3.14159$ ,

so a reasonable estimate of the length of this arc is

$L \approx 30 (3.14159) \approx 94.2477$

Of the five choices, 95 comes closest to the correct length.

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Question

Give the circumference of the circle on the coordinate plane with equation

$x^{2}+y^{2}+ 6x-20y+ 9=0$ .

Answer

We must first rewrite the equation of the circle in standard form

$(x-h)^{2} + (y-k)^{2}= r^{2}$ ;

will be the radius.

$x^{2}+y^{2}+ 6x-20y+ 9=0$

Subtract 9 from both sides, and rearrange the terms remaining on the left as follows:

$x^{2}+y^{2}+ 6x-20y+ 9- 9 =0- 9$

$x^{2}+ 6x + \underline{; ; ; ; ; ; }+y^{2}-20y+ \underline{; ; ; ; ; ; } = - 9$

Note that blanks have been inserted after the linear terms. In these blanks, complete two perfect square trinomials by dividing linear coefficients by 2 and squaring:

$(6 \div 2)^{2} = 3^{2} = 9$

$(20 \div 2)^{2} = 10^{2} = 100$

Add to both sides, as follows:

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = - 9+9+ 100$

$x^{2}+ 6x + 9+y^{2}-20y+ 100 = 100$

Rewrite the expression on the left as the sum of the squares of two binomials.

$(x+3)^{2} + (y-10)^{2} = 100$

The equation is now in standard form.

$r^{2} = 100$

The radius is the square root of this, or

The circumference of a circle is the radius multiplied by $2 \pi$ , so

$C = 2 \pi r = 2 \pi (10)= 20 \pi$ ,

the correct response.

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Question

A circle on the coordinate plane has a diameter with endpoints and . Give its equation.

Answer

The equation, in standard form, of a circle on the coordinate plane with center and radius is

$(x-h)^{2} + (y-k)^{2}= r^{2}$ .

It is necessary to find the center and the radius of the circle in order to determine its equation.

The center of the circle is the midpoint of a given diameter. The midpoint $\left ( x_{M}, y_{M} \right )$ of a segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ can be found by applying the midpoint formula:

$x_{M}= \frac{x_{1}+ x_{2} }{2}$

$y_{M}= \frac{y_{1}+ y_{2} }{2}$

Setting $x_{1}= 3, x_{2}= 9$ :

$x_{M}= \frac{3+9 }{2}= \frac{12}{2} = 6$

Setting $y_{1}= 4, y_{2}= 10$

$y_{M}= \frac{4+10 }{2} = \frac{14}{2} = 7$

The midpoint, and the center of the circle, is at .

The radius of the circle is the distance from the center to an endpoint, so we can use the distance formula

$r = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$ .

Substitute $x_{1}= 6, x_{2}= 9 , y_{1}= 7, y_{2}= 10$ :

$r = \sqrt{(9-6)^{2}+(10-7 )^{2} }$

$= \sqrt{3^{2}+3 ^{2} }$

$= \sqrt{9+9 }$

$= \sqrt{18 }$

and

$r^{2} = 18$

Now, set $r^{2} = 18 , h = 6, k= 7$ in the standard form of the equation of a circle, and the result is

$(x-6)^{2} + (y-7)^{2}= 18$

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Question

A circle on the coordinate plane has a radius with endpoints and . Give its equation.

Answer

The equation, in standard form, of a circle on the coordinate plane with center and radius is

$(x-h)^{2} + (y-k)^{2}= r^{2}$ .

It is necessary to find the center and the radius of the circle in order to determine its equation. The radius of the circle can be determined by applying the distance formula to the coordinates of the endpoints of the radius. However, it is not clear from the information given which endpoint is the center of the circle. Therefore, while can be determined, cannot. The correct response is that insufficient information exists to determine its equation.

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Question

What is the circumference of a circle with an area of $1369\pi$ ?

Answer

Step 1: Since we are given the area, let's first find the radius of the circle.

If the area of a circle is calculated by: $A=\pi r^2$ , we can plug in the values that are given to us. $A=1369\pi$ .

We have $1369\pi=\pi r^2$ . We will divide by $\pi$ .

. To find the radius, we will take the square root of both sides.

So, .

Step 2: Now that we have the radius, we will now find the circumference.

The circumference formula by using the radius is $C=2\pi r$ .

Plug in all the information that we have...

$C=2\pi(37)\Rightarrow C=74\pi$

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Question

A circle on the coordinate plane has its center at and passes through . Give its circumference.

Answer

The radius of the circle is the distance between its center and a point that it passes through. This radius can be calculated using the distance formula:

$r = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$ .

Setting $x_{1} = 2, x_{2} = 10, y_{1}=9, y_{2} =3$ :

$r = \sqrt{(10-2)^{2}+(3-9)^{2} }$

$= \sqrt{8^{2}+(-6)^{2} }$

$= \sqrt{64+36}$

$= \sqrt{100}$

The circumference of a circle is equal to $2 \pi$ multiplied by the radius, so

$C= 2 \pi r = 2 \pi (10) = 20 \pi$ ,

the correct response.

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