Apply concepts of density in modeling situations - 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

The graphic below shows a blueprint for a swimming pool.

If the pool is going to be 66 inches deep, how many cubic feet of water will it be able to hold? (1 ft = 12 in)

Answer

Notice that the outer dimensions of the blueprint are the dimensions for the entire pool, including the concrete, while the inner dimensions are for the part of the pool that will be filled with water. Therefore, we want to focus on just the inner dimensions.

Notice that the depth is given in inches, while the dimensions are in feet. Convert 66 inches to feet by dividing 66 by 12, since 12 inches makes a foot:

$66;in\cdot\frac{1 ft}{12 in}=5.5 ft$

The inch units cancel out and leave us with just the feet units. 66 in is 5.5 ft.

Now we have all of the information we need to solve for the volume of the pool. The pool is a rectangular prism, and the formula for volume of a rectangular prism is

$length\cdot width\cdot height$

(In this case, the "height" of the swimming pool is its depth.)

The blueprint shows that the pool is 40 ft long and 30 ft wide. Plugging in the measurements from the problem, we get

$40,ft\cdot30,ft\cdot5.5,ft$

Multiplying this out, we get .

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