How to find the volume of a net

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ISEE Middle Level Quantitative Reasoning › How to find the volume of a net

Questions 1 - 8

Prism

Give the volume of the above box in cubic centimeters.

$324,000 \textrm{ cm}^{3}$

$32,400 \textrm{ cm}^{3}$

$21,600\textrm{ cm}^{3}$

$28,800 \textrm{ cm}^{3}$

$216,000 \textrm{ cm}^{3}$

Explanation

100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.

$0.9 \times 100 = 90$ centimeters

$0.6 \times 100 = 60$ centimeters

Use the volume formula, substituting :

$V = 90 \cdot 60 \cdot 60 = 324,000$ cubic centimeters

A rectangular prism has length 24 inches, width 18 inches, and height 15 inches. Give its volume in cubic feet.

$3 \frac{3}{4} \textrm{ ft}^{3}$

$4 \frac{3}{4} \textrm{ ft}^{3}$

$4 \frac{1}{4} \textrm{ ft}^{3}$

$4 \frac{1}{2} \textrm{ ft}^{3}$

Explanation

Divide each dimension in inches by 12 to convert from inches to feet:

$L = 24 \div 12 = 2$ feet

$W = 18 \div 12 = 1 \frac{1}{2} = \frac{3}{2}$ feet

$H = 15 \div 12 = 1 \frac{1}{4} = \frac{5}{4}$ feet

Multiply the three to get the volume:

$V = LWH = 2 \cdot \frac{3}{2} \cdot \frac{5}{4} = \frac{15}{4} = 3 \frac{3}{4}$ cubic feet

Swimming_pool

One cubic foot is equal to (about) 7.5 gallons.

The above depicts a rectangular swimming pool for an apartment. The pool is seven feet deep everywhere. How many gallons of water does the pool hold?

$91,875 \textrm{ gal}$

$8,925 \textrm{ gal}$

$89,250 \textrm{ gal}$

$17,850 \textrm{ gal}$

$178,500 \textrm{ gal}$

Explanation

The pool can be seen as a rectangular prism with dimensions 50 feet by 35 feet by 7 feet; its volume in cubic feet the product of these dimensions, which is

$50 \times 35 \times 7 =12,250$ cubic feet.

One cubic foot is equal to about 7.5 gallons, so multiply:

$12,250 \times 7.5 = 91,875$ gallons.

Which is the greater quantity?

(a) The sidelength of a cube with surface area $600 \textrm{ cm}^{2}$

(b) The sidelength of a cube with volume $1,000 \textrm{ cm}^{3}$

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

(a) A cube has six faces, each a square. Since the surface area of this cube is $600 \textrm{ cm}^{2}$ , each face has one-sixth this area, or $100 \textrm{ cm}^{2}$ ; the sidelength is the square root of this, or $10 \textrm{ cm}$ .

(b) The volume of a cube is the cube of its sidelength, so we take the cube root of the volume of this cube to get the sidelength:

$\sqrt[3]{1,000} = 10 \textrm{ cm}$

The cubes have the same sidelength.

Which is the greater quantity?

(a) The volume of a rectangular prism with length 60 centimeters, width 30 centimeters, and height 15 centimeters

(b) The volume of a cube with sidelength 300 millimeters

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

(a) The volume of the prism is the product of its length, its width, and its height:

$V = 60 \times 30 \times 15 = 27,000 \textrm{ cm}^{3}$

(b) The volume of the cube is the cube of its sidelength. We restate 300 millimeters as 30 centimeters, and cube this:

$V = 30 ^{3} = 30 \times 30 \times 30 = 27,000 \textrm{ cm}^{3}$

The volumes are equal.

Prism

Give the volume of the box in cubic inches.

$3,168 \textrm{ in}^{3}$

$1,056 \textrm{ in}^{3}$

$672 \textrm{ in}^{3}$

$816 \textrm{ in}^{3}$

$1,344 \textrm{ in}^{3}$

Explanation

Use the volume formula, substituting :

$V=22 \cdot 12 \cdot 12 = 3168^{3}$

Swimming_pool

The above depicts a rectangular swimming pool for an apartment. The pool is six feet deep everywhere. What is the volume of the pool in cubic feet?

$10,500\textrm{ ft}^{3}$

$9,500\textrm{ ft}^{3}$

$510\textrm{ ft}^{3}$

$1,020\textrm{ ft}^{3}$

$51,000\textrm{ ft}^{3}$

Explanation

The pool can be seen as a rectangular prism with dimensions 50 feet by 35 feet by 6 feet; its volume is

$50 \times 35 \times 6 = 10,500$ cubic feet.

A cube made of nickel has sidelength 20 centimeters, Nickel has a density of 8.9 grams per cubic centimeter. What is the mass of this cube?

$71.2 \textrm{ kg}$

$7.12 \textrm{ kg}$

$35.6 \textrm{ kg}$

$3.56 \textrm{ kg}$

Explanation

The volume of the cube is $20 ^{3} = 20 \times 20 \times 20 = 8,000$ cubic centimeters. Multiply by the number of grams per cubic centimeter to get $8,000 \times 8.9 = 71,200$ grams, or $71,200 \div 1,000 = 71.2$ kilograms.

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