ISEE Middle Level Quantitative Reasoning - Solid Geometry

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

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

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

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.

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.

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

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.

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

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 $10 \textrm{ cm}$ .