Math 2 Quiz: Volume And Scale Factors

14 questions · exam conditions

0:00

Volume And Scale FactorsQuestion 1 of 14

A company manufactures similar cube-shaped containers in two sizes. The larger container holds $8$ times as much volume as the smaller one. If the smaller container uses $24$ square inches of material for its surface, how much material does the larger container use?

48

square inches

96

square inches

144

square inches

192

square inches

← Back to quizzes

Math 2 Quiz

Math 2 Quiz: Volume And Scale Factors

Practice Volume And Scale Factors in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Volume And Scale Factors, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$48$ square inches
$96$ square inches (correct answer)
$144$ square inches
$192$ square inches

Explanation: If the volume ratio is 8:1, then the linear dimension ratio is ∛8:1 = 2:1. Surface area scales as the square of linear dimensions, so the surface area ratio is 2²:1 = 4:1. Therefore, the larger container uses 24 × 4 = 96 square inches of material. Choice A doubles instead of using the area scaling. Choice C incorrectly uses the relationship 24 × 6 = 144. Choice D uses the volume ratio directly instead of the area relationship.

Question 2

A jewelry designer creates similar pendant designs in different sizes. The volume of gold needed for the large pendant is $2.197$ times the volume needed for the small pendant. If the small pendant has a longest dimension of $18$ mm, what is the longest dimension of the large pendant?

$22.2$ mm
$24.0$ mm (correct answer)
$27.6$ mm
$39.5$ mm

Explanation: Since the volume ratio is 2.197, the linear scale factor is ∛2.197 ≈ 1.3 (since 1.3³ = 2.197). Therefore, the longest dimension of the large pendant is 18 × 1.3 = 23.4 mm, which rounds to 24.0 mm. Choice A uses an incorrect cube root calculation. Choice C uses 18 × 1.533 ≈ 27.6, which comes from incorrectly taking 2.197^(2/3). Choice D uses approximately 18 × 2.197 = 39.5, applying the volume ratio directly to the linear dimension.

Question 3

A spherical tank is replaced by a similar spherical tank with $3$ times the diameter. If the original tank held $500$ gallons when full, how many gallons does the new tank hold when it is $\frac{2}{3}$ full?

$4500$ gallons
$9000$ gallons (correct answer)
$13500$ gallons
$18000$ gallons

Explanation: When diameter increases by a factor of 3, volume increases by a factor of 3³ = 27. The new tank's full capacity is 500 × 27 = 13,500 gallons. When 2/3 full, it holds (2/3) × 13,500 = 9,000 gallons. Choice A uses an incorrect volume scaling. Choice C gives the full capacity of the new tank. Choice D incorrectly applies the scaling to the partial volume differently.

Question 4

Two similar spherical balloons are inflated with helium. The larger balloon has a diameter that is $40\%$ larger than the smaller balloon. If the smaller balloon contains $2$ cubic feet of helium, how much helium does the larger balloon contain?

$2.8$ cubic feet
$3.92$ cubic feet
$8.0$ cubic feet
$5.49$ cubic feet (correct answer)

Explanation: When you encounter problems involving similar three-dimensional shapes with different sizes, remember that volume scales with the cube of the linear dimension ratio. This is because volume involves three dimensions multiplied together. Since the larger balloon's diameter is 40% larger than the smaller balloon's, the diameter ratio is

1.4:1

(or simply

1.4

). For similar spheres, all corresponding linear dimensions scale by this same ratio. However, volume scales by the cube of this ratio. So the volume ratio is

1.4^3 = 2.744

. Since the smaller balloon contains

2

cubic feet of helium, the larger balloon contains

2 \times 2.744 = 5.488

cubic feet, which rounds to

5.49

cubic feet. Choice A (

2.8

cubic feet) represents the common mistake of using the linear scaling factor directly:

2 \times 1.4 = 2.8

. This ignores that volume is three-dimensional. Choice B (

3.92

cubic feet) comes from incorrectly using the square of the linear ratio:

2 \times 1.4^2 = 2 \times 1.96 = 3.92

. This would be correct for area scaling, but not volume. Choice C (

8.0

cubic feet) suggests using

2^3 = 8

, perhaps confusing the volume scaling concept by cubing the original volume instead of the size ratio. Study tip: For similar 3D shapes, always remember the

1:2:3

rule—linear dimensions scale by the ratio, areas by the ratio squared, and volumes by the ratio cubed. Don't fall for the trap of applying linear scaling to volume problems.

Question 5

A perfume manufacturer produces similar bottle designs in different sizes. The medium bottle has $1.5$ times the height of the small bottle and contains $60$ mL of perfume. How much perfume does the small bottle contain?

$17.8$ mL (correct answer)
$26.7$ mL
$40$ mL
$44.4$ mL

Explanation: If the medium bottle has 1.5 times the height of the small bottle, then all linear dimensions are scaled by 1.5 (since the bottles are similar). Volume scales as the cube of linear dimensions, so the medium bottle has volume 1.5³ = 3.375 times that of the small bottle. If the medium bottle contains 60 mL, then the small bottle contains 60/3.375 = 17.78 ≈ 17.8 mL. Choice B uses the wrong power (60/1.5² = 26.7). Choice C uses direct proportion (60/1.5 = 40). Choice D results from an incorrect calculation.

Question 6

Three similar rectangular prisms have volumes in the ratio $8:27:64$ . If the middle-sized prism has a surface area of $54$ square units, what is the surface area of the largest prism?

$72$ square units
$96$ square units (correct answer)
$108$ square units
$144$ square units

Explanation: The volume ratios

8:27:64

correspond to linear scale factors of

2:3:4

(since

2^3=8

3^3=27

4^3=64

). Surface area scales as the square of the linear scale factor. The ratio of surface areas is

2^2:3^2:4^2 = 4:9:16

. If the middle prism has surface area

54

, then the largest has surface area

54 \times \frac{16}{9} = 96

square units.

Question 7

A scale model of a building is constructed where $1$ inch on the model represents $8$ feet on the actual building. If a room in the actual building has a volume of $12,288$ cubic feet, what is the volume of the corresponding room in the scale model?

$1.5$ cubic inches
$12$ cubic inches
$24$ cubic inches (correct answer)
$96$ cubic inches

Explanation: The linear scale factor from actual to model is

\frac{1 \text{ inch}}{8 \text{ feet}} = \frac{1}{96}

(converting feet to inches). The volume scale factor is

(1/96)^3 = \frac{1}{884,736}

. The model room volume is

12,288 \times \frac{1}{884,736} = \frac{12,288}{884,736} = 24

cubic inches.

Question 8

A jewelry designer creates similar triangular pyramid pendants. The larger pendant requires $3.375$ times as much gold by volume as the smaller pendant. If the smaller pendant has a base edge length of $6$ mm, what is the base edge length of the larger pendant?

$9$ mm (correct answer)
$10.5$ mm
$12$ mm
$15$ mm

Explanation: Since volume scales as the cube of the linear scale factor, if the volume ratio is

3.375

, then the linear scale factor is

\sqrt[3]{3.375} = 1.5

. The base edge length of the larger pendant is

6 \times 1.5 = 9

mm.

Question 9

A company manufactures similar spherical storage containers in two sizes. The surface area of the larger container is $2.25$ times the surface area of the smaller container. If the smaller container has a volume of $288\pi$ cubic feet, what is the volume of the larger container?

$432\pi$ cubic feet
$648\pi$ cubic feet
$972\pi$ cubic feet (correct answer)
$1458\pi$ cubic feet

Explanation: Since surface area scales as the square of the linear scale factor, if the surface area ratio is

2.25 = (1.5)^2

, then the linear scale factor is

1.5

. Volume scales as the cube of the linear scale factor, so the volume ratio is

(1.5)^3 = 3.375

. The larger container's volume is

288\pi \times 3.375 = 972\pi

cubic feet.

Question 10

A manufacturer produces similar hexagonal prism containers. When the linear dimensions are increased by a factor of $k$ , the new container costs $12.60 more to fill with liquid than the original container. If the liquid costs $0.35 per cubic unit and $ $k = 1.4$ $, what is the volume of the original container?

$12$ cubic units
$21$ cubic units
$18$ cubic units
$15$ cubic units (correct answer)

Explanation: When linear dimensions of a 3D object are scaled by factor

k

, the volume scales by

k^3

. This is because volume involves three dimensions (length × width × height), so each dimension multiplied by

k

gives us

k \times k \times k = k^3

. Let's call the original volume

V

. After scaling by

k = 1.4

, the new volume becomes

V \times (1.4)^3 = V \times 2.744

. The increase in volume is

2.744V - V = 1.744V

. Since this extra volume costs $12.60 to fill at $0.35 per cubic unit, we can set up the equation: $

1.744V \times 0.35 = 12.60

$ Solving for

1.744V = \frac{12.60}{0.35} = 36

V = \frac{36}{1.744} = \frac{36000}{1744} = \frac{2250}{109.} \approx 15

Answer D (

cubic units) is correct. Looking at the wrong answers: Answer A (

cubic units) might come from incorrectly using

k^2

instead of

k^3

for volume scaling. Answer B (

cubic units) could result from computational errors in the division. Answer C (

cubic units) might arise from rounding errors or mishandling the scaling factor calculations. Study tip: Remember that scaling relationships depend on dimensionality—linear measurements scale by

, areas by

k^2

, and volumes by

k^3

. Always identify what type of quantity you're working with before applying scaling factors.

Question 11

Two similar cylinders have surface areas in the ratio $4:9$ . If the volume of the smaller cylinder is $200$ cubic centimeters, what is the volume of the larger cylinder?

$300$ cubic centimeters
$450$ cubic centimeters
$675$ cubic centimeters (correct answer)
$1800$ cubic centimeters

Explanation: For similar solids, surface area ratios equal the square of linear dimension ratios, and volume ratios equal the cube of linear dimension ratios. If surface areas are in ratio 4:9, then linear dimensions are in ratio √4:√9 = 2:3. Therefore, volumes are in ratio 2³:3³ = 8:27. If the smaller volume is 200, then 200/V = 8/27, so V = 200 × 27/8 = 675. Choice A incorrectly uses the surface area ratio directly. Choice B uses an incorrect intermediate calculation. Choice D uses 9² instead of the proper volume relationship.

Question 12

A spherical balloon is inflated so that its radius increases from $6$ inches to $9$ inches. By what factor does the volume increase?

$1.5$
$2.25$
$4.5$
$3.375$ (correct answer)

Explanation: When you encounter questions about how volume changes as dimensions change, you're dealing with scaling relationships. The key insight is that volume scales with the cube of linear dimensions. The volume of a sphere is

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

. To find how volume changes, you need to compare the volumes at the two different radii. Initial volume:

V_1 = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216)

Final volume:

V_2 = \frac{4}{3}\pi (9)^3 = \frac{4}{3}\pi (729)

The factor by which volume increases is:

\frac{V_2}{V_1} = \frac{729}{216} = \frac{9^3}{6^3} = \left(\frac{9}{6}\right)^3 = (1.5)^3 = 3.375

Choice A (

1.5

) represents the linear scaling factor—how much the radius increased. This is a common trap since students sometimes forget that volume scales cubically, not linearly. Choice B (

2.25

) equals

(1.5)^2

, which would be correct for area scaling. This catches students who remember that area scales with the square but apply it incorrectly to volume. Choice C (

4.5

) equals

1.5 \times 3

, suggesting someone multiplied the linear factor by 3 (perhaps thinking "three dimensions"). This reflects confusion about how dimensional scaling works. Choice D (

3.375

) correctly applies the cubic relationship:

(1.5)^3

. Remember this pattern: when any linear dimension of a 3D object scales by factor

k

, the volume scales by

k^3

. Always cube the linear scaling factor for volume problems.

Question 13

Two similar pyramids have volumes of $54$ cubic feet and $128$ cubic feet respectively. If the height of the smaller pyramid is $9$ feet, what is the height of the larger pyramid?

$12$ feet (correct answer)
$14$ feet
$16$ feet
$18$ feet

Explanation: The volume ratio is 128/54 = 64/27. Since volume scales as the cube of linear dimensions, the linear scale factor is ∛(64/27) = ∛64/∛27 = 4/3. Therefore, the height of the larger pyramid is 9 × (4/3) = 12 feet. Choice B might result from approximating the cube root incorrectly. Choice C assumes a scale factor of 16/9 from direct proportion. Choice D uses double the original height.

Question 14

Two similar rectangular prisms have corresponding edge lengths in the ratio $3:5$ . If the smaller prism has a volume of $162$ cubic units, what is the volume of the larger prism?

$270$ cubic units
$450$ cubic units
$750$ cubic units (correct answer)
$1250$ cubic units

Explanation: For similar solids, the ratio of volumes equals the cube of the ratio of corresponding linear dimensions. The linear ratio is 3:5, so the volume ratio is 3³:5³ = 27:125. If the smaller volume is 162, then 162/V = 27/125, so V = 162 × 125/27 = 750. Choice A uses the linear ratio instead of cubing it. Choice B incorrectly squares the ratio. Choice D cubes the ratio but applies it in the wrong direction.