The volume of a cone, given radius $\displaystyle r$ and height $\displaystyle h$, can be calculated using the formula

$\displaystyle V = \frac{1}{3} \pi r^{2}h$.

We are given that $\displaystyle r = 10$, but we are not given the value of $\displaystyle h$. We are given slant height $\displaystyle l = 20$, and since the cone is a right cone, its radius, height, and slant height can be related using the Pythagorean relation

$\displaystyle r^{2}+ h^{2}= l^{2}$.

Substituting 10 for $\displaystyle r$ and 20 for $\displaystyle l$, we can find $\displaystyle h$:

$\displaystyle 10^{2}+ h^{2}= 20^{2}$

$\displaystyle 100+ h^{2}=400$

$\displaystyle 100+ h^{2} - 100 =400 - 100$

$\displaystyle h^{2} =300$

$\displaystyle h = \sqrt{300 }$

Now substitute in the volume formula:

$\displaystyle V = \frac{1}{3} \pi r^{2}h$

$\displaystyle = \frac{1}{3} \pi (10^{2}) \sqrt{300}$

$\displaystyle \approx \frac{1}{3} \cdot 3.1416 \cdot 100 \cdot 17.3$

$\displaystyle \approx 1813.8$

To the nearest whole, this is 1,814.

The volume of a cone is given by the formula 

$\displaystyle V= \frac{1}{3}\pi r^2h$

We are given that $\displaystyle h=1$. Thus, the volume of our cone is given by

$\displaystyle V= \frac{1}{3}\pi r^2$.

We are given that the volume of our cone is $\displaystyle 12.56$.

Thus,

$\displaystyle 12.56=\frac{1}{3}\pi r^2$

so

$\displaystyle \frac{12.56 \times3}{\pi} = r^2$

and

$\displaystyle 12 = r^2$.

Thus, 

$\displaystyle \sqrt{12}=r$.

so the correct answer is $\displaystyle \sqrt{12}\;cm$.

The surface area of a sphere can be calculated using the formula

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

Solving for $\displaystyle r$:

Set $\displaystyle A = 100 \pi$ and divide both sides by $\displaystyle 4 \pi$:

$\displaystyle 4 \pi r^{2} = 100 \pi$

$\displaystyle \frac{4 \pi r^{2}}{4 \pi} = \frac{100 \pi}{4 \pi}$

$\displaystyle r^{2} =25$

Take the square root of both sides:

$\displaystyle r = \sqrt{25} = 5$

Set $\displaystyle r=5$ in the volume formula:

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

$\displaystyle V= \frac{4}{3} \pi \left (5 \right )^{3} = \frac{4}{3} \pi \left (125 \right ) = \frac{500}{3}\pi$,

the correct response.

The surface area $\displaystyle A$ of a sphere, given its radius $\displaystyle r$, is equal to

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

Substitute $\displaystyle 64\pi$ for $\displaystyle A$ and solve for $\displaystyle r$:

$\displaystyle 4 \pi r^{2} = 64 \pi$

Divide by $\displaystyle 4 \pi$:

$\displaystyle \frac{4 \pi r^{2}}{4 \pi} = \frac{64 \pi}{4 \pi}$

$\displaystyle r^{2} = 16$

Extract the square root:

$\displaystyle r= 4$

Substitute for $\displaystyle r$ in the volume formula:

$\displaystyle V = \frac{4}{3} \pi r^{3}= \frac{4}{3} \pi \cdot 4^{3} =\frac{4}{3} \pi \cdot 64 = \frac{256}{3}\pi$,

the correct response.

A square has area formula

 $\displaystyle A = l^2$

The total area required for the chickens will be

$\displaystyle 25 \;\textup{m}^2$

since each chicken requires $\displaystyle 1\;\textup{m}^2$ of space and there are $\displaystyle 25$ chickens.

Thus, we have

$\displaystyle 25 = l^2$

for the length of our chicken fence.

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

$\displaystyle l = 5$

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

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

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

Alabama:

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

Arkansas:

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

Kentucky:

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

Mississippi:

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

Tennessee:

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

Tennessee has the greatest population density among the five states. 

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:

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

North Dakota:

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

South Dakota:

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

Vermont:

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

Wyoming:

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

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

The county is in the shape of a trapezoid with bases of length $\displaystyle B = 44$ and $\displaystyle b= 33$, and with height $\displaystyle h = 30$. Its area in square miles can be found by substituting in the formula for the area of a trapezoid:

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

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

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

$\displaystyle A = 1,155$ square miles

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

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

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

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:

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

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

$\displaystyle length\cdot width\cdot height$

(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

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

Multiplying this out, we get $\displaystyle 6,600\,ft^3$.

On the horizontal axis of a histogram, you have labels representing ranges of values. For example, the label "140–159" represents the range of islands with populations between 140 and 159. 

The question asks for the percent of islands with populations under 140. The ranges that are smaller than 140 are the "120–139" range and the "100–119" range. Match the bars of each with the percents on the vertical (y) axis. The percents corresponding to these ranges are 35% and 30%, respectively.

Adding both of these percents yields the answer, 65%.