Volume

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SAT Subject Test in Math II › Volume

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A circular swimming pool has diameter 40 meters and depth meters throughout. Which of the following expressions gives the amount of water it holds, in cubic meters?

$400 \pi t$

CORRECT

$1,600 \pi t$

$40 \pi t$

$80 \pi t$

$160 \pi t$

Explanation

The pool can be seen as a cylinder with depth (or height) , and a base with diameter 40 m - and radius half this, or . The capacity of the pool is the volume of this cylinder, which is

$V = \pi r^{2}h = \pi \cdot 20^{2}\cdot t = 400 \pi t$

The bottom surface of a rectangular prism has area 100; the right surface has area 200; the rear surface has area 300. Give the volume of the prism (nearest whole unit), if applicable.

CORRECT

Explanation

Let the dimensions of the prism be , , and .

Then, , , and .

From the first and last equations, dividing both sides, we get

$\frac{WH}{LW} = \frac{300}{100}$

$\frac{H}{L} = 3$

Along with the second equation, multiply both sides:

$\frac{H}{L} \cdot LH = 3 \cdot 200$

$H ^{2}= 600$

Taking the square root of both sides and simplifying, we get

$H = \sqrt{600} = \sqrt{100} \cdot \sqrt{6} = 10 \sqrt{6}$

Now, substituting and solving for the other two dimensions:

$L \cdot 10 \sqrt{6} = 200$

$L = \frac{200}{10 \sqrt{6}} = \frac{20}{\sqrt{6}}$

$W \cdot 10 \sqrt{6} = 300$

$W = \frac{300}{10 \sqrt{6}} = \frac{30 }{\sqrt{6}}$

Now, multiply the three dimensions to obtain the volume:

$= \frac{20}{\sqrt{6}} \cdot 10 \sqrt{6} \cdot \frac{30 }{\sqrt{6}}$

$= \frac{20}{\sqrt{6}} \cdot\frac{ 10 \sqrt{6}}{1} \cdot \frac{30 }{\sqrt{6}}$

$= \frac{ 6,000 \sqrt{6}} {6}$

$= 1,000 \sqrt{6}$

$\approx 1,000 \cdot 2.449$

$\approx 2,449$

The radius and the height of a cylinder are equal. If the volume of the cylinder is , what is the diameter of the cylinder?

CORRECT

Explanation

Recall how to find the volume of a cylinder:

$\text{Volume}=\pi r^2 h$

Since we know that the radius and the height are equal, we can rewrite the equation:

$\text{Volume}=\pi r^2 (r)=\pi r^3$

Using the given volume, find the length of the radius.

$13=\pi r^3$

Since the question asks you to find the diameter, multiply the radius by two.

The width of a box is two-thirds its height and three-fifths its length. The volume of the box is 6 cubic meters. To the nearest centimeter, give the width of the box.

$134 \textrm{ cm}$

CORRECT

$201 \textrm{ cm}$

$223 \textrm{ cm}$

$246 \textrm{ cm}$

$164 \textrm{ cm}$

Explanation

Call , , and the length, width, and height of the crate.

The width is two-thirds the height, so

$W = \frac{2}{3} H$ .

Equivalently,

$\frac{3} {2} \cdot \frac{2}{3} H = \frac{3} {2} W$

$H = \frac{3} {2} W$

The width is three-fifths the length, so

$W = \frac{3}{5} L$ .

Equivalently,

$\frac{5} {3} \cdot \frac{3}{5} L = \frac{5} {3} \cdot W$

$L = \frac{5} {3 } W$

The dimensions of the crate in terms of are , $\frac{3} {2} W$ , and $\frac{5} {3 } W$ . The volume is their product:

Substitute:

$\frac{5} {3 } W \cdot W \cdot \frac{3} {2} W = V$

$\frac{5} {3 } \cdot \frac{3} {2} \cdot W \cdot W \cdot W = 6$

$\frac{5} {2} \cdot W ^{3}= 6$

$\frac{2} {5} \cdot \frac{5} {2} \cdot W ^{3}= \frac{2} {5} \cdot 6$

$W ^{3}= \frac{12} {5}$

Taking the cube root of both sides:

$W =\sqrt[3]{ \frac{12} {5}}$

$W \approx 1.339$ meters.

Since one meter comprises 100 centimeters, multiply by 100 to convert to centimeters:

$1.339 \cdot 100 = 133.9$ centimeters,

which rounds to 134 centimeters.

One cubic meter is equal to one thousand liters.

A circular swimming pool is meters in diameter and meters deep throughout. How many liters of water does it hold?

$250\pi d^{2}f$

CORRECT

$500 \pi d^{2}f$

$1,000\pi d^{2}f$

$1,000\pi df$

$500\pi df$

Explanation

The pool can be seen as a cylinder with depth (or height) , and a base with diameter - and, subsequently, radius half this, or $\frac{d}{2}$ . The volume of the pool in cubic meters is

$V = \pi r^{2}h = \pi \left ( \frac{d}{2} \right )^{2} \cdot f = \pi \cdot \frac{d^{2}}{4} \cdot f = \frac{ \pi d^{2}f}{4}$

Multiply this number of cubic meters by 1,000 liters per cubic meter:

$V = \frac{ \pi d^{2}f}{4} \cdot \1,000 = 250\pi d^{2}f$

Pool

The above depicts a rectangular swimming pool for an apartment.

On the left and right edges, the pool is three feet deep; the dashed line at the very center represents the line along which it is eight feet deep. Going from the left to the center, its depth increases uniformly; going from the center to the right, its depth decreases uniformly.

In cubic feet, how much water does the pool hold?

$9,625 \textrm{ ft}^{3}$

CORRECT

$9,875 \textrm{ ft}^{3}$

$10,225 \textrm{ ft}^{3}$

$1,870 \textrm{ ft}^{3}$

$2,070 \textrm{ ft}^{3}$

Explanation

The pool can be looked at as a pentagonal prism with "height" 35 feet and its bases the following shape (depth exaggerated):

Pool

This is a composite of two trapezoids, each with bases 3 feet and 8 feet and height 25 feet; the area of each is

$A = \frac{1}{2} (3+8) \cdot 25 = 137.5$ square feet.

The area of the base is twice this, or

$B = 137.5 \cdot 2 = 275$ square feet.

The volume of a prism is its height times the area of its base, or

$V = 275 \cdot 35 = 9,625$ cubic feet, the capacity of the pool.

One cubic meter is equal to one thousand liters.

A rectangular swimming pool is meters deep throughout and meters wide. Its length is ten meters greater than twice its width. How many liters of water does the pool hold?

$2,000 DW^{2} + 10,000 DW$

CORRECT

$2,000 DW^{2} -10,000 DW$

$0.002 DW^{2} + 0.01 DW$

$0.002 DW^{2} - 0.01 DW$

None of the other responses is correct.

Explanation

Since the length of the pool is ten meters longer than twice its width , its length is .

The inside of the pool can be seen as a rectangular prism, and as such, its volume in cubic feet can be calculated as the product of its length, width, and height (or depth). This product is

$V = D \cdot (2W + 10) \cdot W$

$V = 2DW^{2} + 10DW$

Multiply this by the conversion factor 1,000, and its volume in liters is

$V =1,000 \left ( 2DW^{2} + 10DW \right )$

$V = 2,000 DW^{2} + 10,000 DW$

Find the volume of a sphere with a diameter of 10.

$166.67\pi$

CORRECT

$100\pi$

$1333.33\pi$

$25\pi$

$33.33\pi$

Explanation

The surface area of a sphere is found using the formula $A=\frac{4}{3}\pi r^3$ . We are given the diameter of the circle and so we have to use it to find the radius (r).