Math 2 Quiz: Dimension Changes And Volume

19 questions · exam conditions

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Dimension Changes And VolumeQuestion 1 of 19

A rectangular prism has dimensions 4 cm × 6 cm × 8 cm. If each dimension is increased by the same amount $x$ , and the new volume is exactly double the original volume, what is the value of $x$ ?

1.26 cm

2.00 cm

2.52 cm

4.00 cm

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Math 2 Quiz

Math 2 Quiz: Dimension Changes And Volume

Practice Dimension Changes And Volume 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 Dimension Changes And Volume, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

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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.

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Question 1

A rectangular prism has dimensions 4 cm × 6 cm × 8 cm. If each dimension is increased by the same amount $x$ , and the new volume is exactly double the original volume, what is the value of $x$ ?

1.26 cm (correct answer)
2.00 cm
2.52 cm
4.00 cm

Explanation: Original volume = 4 × 6 × 8 = 192 cm³. New volume = (4+x)(6+x)(8+x) = 384 cm³. Expanding: (4+x)(6+x)(8+x) = 192 + 72x + 18x² + x³ = 384. This gives x³ + 18x² + 72x - 192 = 0. Solving this cubic equation yields x ≈ 1.26. Choice B assumes uniform scaling by factor 2, giving x = 4. Choice C doubles choice A. Choice D comes from incorrectly thinking each dimension doubles.

Question 2

Two cylindrical silos are geometrically similar. The smaller silo has radius $4$ feet and stores $1200$ cubic feet of grain. If the larger silo has radius $10$ feet, how much grain can it store?

$3000 \text{ cubic feet}$
$7500 \text{ cubic feet}$
$15625 \text{ cubic feet}$
$18750 \text{ cubic feet}$ (correct answer)

Explanation: The radius ratio is 10:4 = 2.5:1. For similar cylinders, the volume ratio is (2.5)³ = 15.625. Therefore, the larger silo stores 1200 × 15.625 = 18,750 cubic feet. Choice A uses ratio 10/4. Choice B uses ratio (10/4)². Choice C uses 15.625 × 1000 instead of × 1200.

Question 3

A manufacturer produces conical containers. Due to material constraints, they need to reduce the radius by $20\%$ while keeping the same volume. By approximately what percentage must they increase the height?

$25\%$
$56\%$ (correct answer)
$64\%$
$80\%$

Explanation: For a cone, V = (1/3)πr²h. If radius decreases by 20%, the new radius is 0.8r, so r² becomes (0.8)²r² = 0.64r². To maintain the same volume, height must increase by factor 1/0.64 = 1.5625, which is a 56.25% increase. Choice A uses 1/0.8 - 1. Choice C uses (1/0.8)² - 1. Choice D incorrectly uses the radius reduction percentage.

Question 4

A cylindrical tank with radius $r$ and height $h$ is redesigned so that both the radius and height are scaled by a factor of $k$ . If the original tank holds 500 gallons and the new tank holds 4000 gallons, what scaling factor $k$ was used?

$k = 2$ (correct answer)
$k = 4$
$k = 8$
$k = \sqrt[3]{8}$

Explanation: When both radius and height are scaled by factor k, the volume scales by k³. Since volume increased from 500 to 4000 gallons, we have k³ = 4000/500 = 8, so k = ∛8 = 2. Choice B incorrectly uses k² scaling. Choice C uses the volume ratio directly. Choice D gives the cube root without simplifying.

Question 5

A rectangular prism has dimensions $2x$ by $3x$ by $4x$ . If each dimension is increased by the same amount $d$ , and the resulting volume is exactly twice the original volume, which equation correctly models this situation?

$(2x + d)(3x + d)(4x + d) = 2(24x^3)$ (correct answer)
$(2x + d)(3x + d)(4x + d) = 24x^3 + d^3$
$(2 + d)(3 + d)(4 + d) = 2(24)$
$24x^3 + d = 2(24x^3)$

Explanation: The original volume is (2x)(3x)(4x) = 24x³. After adding d to each dimension, the new volume is (2x+d)(3x+d)(4x+d), which must equal twice the original volume: 2(24x³). Choice B incorrectly adds d³ instead of doubling. Choice C removes the scaling factor x. Choice D incorrectly models the dimension change as adding d to volume.

Question 6

Two similar pyramids have corresponding edge lengths in the ratio $3:5$ . If the smaller pyramid has a volume of $162 \text{ cm}^3$ , and material costs $0.08 per cubic centimeter, what is the cost to fill the larger pyramid with the same material?

$37.80
$60.48 (correct answer)
$75.00
$100.80

Explanation: Since the pyramids are similar with edge ratio 3:5, their volume ratio is (3/5)³ = 27/125. If the smaller volume is 162 cm³, then the larger volume is 162 × (125/27) = 162 × 125/27 = 750 cm³. Cost = 750 × $0.08 = $60.48. Choice A uses ratio 5/3 instead of (5/3)³. Choice C incorrectly uses ratio 5/3. Choice D uses the sum of volumes.

Question 7

A hemispherical bowl has radius $r$ . A new bowl is designed with radius $1.5r$ . If the original bowl holds $2$ liters when full, and both bowls are filled to the same fraction of their capacity, how many more liters does the larger bowl hold than the smaller bowl?

$1.75 \text{ liters}$
$2.50 \text{ liters}$
$4.75 \text{ liters}$ (correct answer)
$6.00 \text{ liters}$

Explanation: The volume scales as the cube of the radius ratio. With radius ratio 1.5:1, the volume ratio is (1.5)³ = 3.375. The larger bowl holds 2 × 3.375 = 6.75 liters when full. The difference is 6.75 - 2 = 4.75 liters. Choice A uses ratio (1.5)² - 1. Choice B uses ratio 1.5 - 0.5. Choice D gives the larger bowl's capacity, not the difference.

Question 8

A water tank shaped like a rectangular prism is $75\%$ full. If all three dimensions of the tank are increased by the same factor $k$ , and the same amount of water now fills only $12\%$ of the new tank, what is the value of $k$ ?

$k = 2.5$ (correct answer)
$k = 4$
$k = 6.25$
$k = \sqrt{6.25}$

Explanation: Let original tank volume be V. Water amount is 0.75V. New tank volume is k³V. The equation is 0.75V = 0.12(k³V), so 0.75 = 0.12k³, giving k³ = 0.75/0.12 = 6.25, so k = ∛6.25 = 2.5. Choice B uses k² scaling. Choice C uses the value of k³. Choice D gives √6.25 instead of ∛6.25.

Question 9

A cylindrical water tank has its radius increased by 20% while its height remains constant. If the original tank held 1000 gallons, what is the new capacity?

1200 gallons
1440 gallons (correct answer)
1728 gallons
2000 gallons

Explanation: Volume of a cylinder is V = πr²h. When radius increases by 20%, the new radius is 1.2r. The new volume is π(1.2r)²h = π(1.44r²)h = 1.44πr²h = 1.44 times the original volume. Therefore: 1.44 × 1000 = 1440 gallons. Choice A incorrectly multiplies by 1.2 instead of 1.2². Choice C uses 1.2³ = 1.728, incorrectly treating this as a 3D scaling. Choice D doubles the volume, perhaps confusing 20% increase with 100% increase.

Question 10

A cube is scaled so that each edge length is multiplied by a factor of $k$ . If the volume increases by 125%, what is the value of $k$ ?

1.25
1.44
1.50 (correct answer)
2.25

Explanation: If volume increases by 125%, the new volume is 225% of the original, or 2.25 times the original. Since volume scales as k³, we have k³ = 2.25. Therefore k = ∛(2.25) = ∛(9/4) = 1.5. Choice A incorrectly uses the percentage increase directly. Choice B might come from squaring 1.2. Choice D is the volume scaling factor itself, not the linear scaling factor.

Question 11

A sphere of radius $r$ is inscribed in a cube. If the cube's edge length is doubled, by what factor does the volume of the inscribed sphere increase?

2
4
6
8 (correct answer)

Explanation: When a sphere is inscribed in a cube, the sphere's diameter equals the cube's edge length, so the sphere's radius is half the edge length. If the cube's edge doubles, the sphere's radius also doubles. Since volume scales as the cube of linear dimensions, the sphere's volume increases by a factor of 2³ = 8. Choice A is the linear scaling factor. Choice B is the area scaling factor. Choice C might come from confusing surface area scaling with an additional factor.

Question 12

Two similar cones have volumes in the ratio 8:27. If the smaller cone has a base radius of 6 cm, what is the base radius of the larger cone?

8 cm
9 cm (correct answer)
12 cm
18 cm

Explanation: For similar solids, the ratio of volumes equals the cube of the ratio of corresponding linear dimensions. If the volume ratio is 8:27, then the linear ratio is ∛(8):∛(27) = 2:3. Since the smaller radius is 6 cm, the larger radius is 6 × (3/2) = 9 cm. Choice A uses the cube root of 8 incorrectly. Choice C doubles the smaller radius. Choice D triples the smaller radius, using the volume ratio instead of the linear ratio.

Question 13

A water tank in the shape of a truncated cone (frustum) has its top radius, bottom radius, and height all increased by 25%. What is the percentage increase in volume?

25% increase from linear dimension scaling
75% increase from additive dimensional effects
95.3% increase using cubic scaling principles (correct answer)
125% increase through area scaling methods

Explanation: When all linear dimensions of any 3D object are scaled by the same factor, volume scales by the cube of that factor. Here, scaling factor is 1.25, so volume scaling is (1.25)³ = 1.953125, which means the new volume is about 195.3% of the original, representing a 95.3% increase. Choice A uses only the linear scaling percentage. Choice B might come from 3×25%. Choice D incorrectly uses area scaling (1.25)² - 1 = 56.25%, then rounds to 125%.

Question 14

A cube is inscribed in a sphere of radius $R$ . If the sphere's radius increases to $1.5R$ , by what factor does the volume of the inscribed cube increase?

1.5 times larger through direct radius scaling
2.25 times larger using surface area relationships
4.5 times larger via geometric constraint analysis
3.375 times larger from cubic scaling effects (correct answer)

Explanation: When you see problems involving inscribed shapes and scaling, the key insight is understanding how linear scaling affects different geometric properties. A cube inscribed in a sphere means all vertices of the cube touch the sphere's surface. For a cube inscribed in a sphere, the sphere's diameter equals the cube's space diagonal (the diagonal from one corner to the opposite corner). If the cube has side length

s

, its space diagonal is

s\sqrt{3}

. Therefore, the sphere's diameter is

s\sqrt{3}

, making the radius

\frac{s\sqrt{3}}{2}

. When the radius increases from

R

1.5R

, the scaling factor is 1.5. Since the cube remains inscribed, its space diagonal must also scale by 1.5, which means each side length scales by 1.5. The new side length becomes

1.5s

. Volume scales as the cube of the linear scaling factor. The original volume was

s^3

, and the new volume is

(1.5s)^3 = 1.5^3 \cdot s^3 = 3.375s^3

. The volume increases by a factor of 3.375. Answer A incorrectly applies linear scaling to volume. Answer B uses

1.5^2 = 2.25

, which would be correct for area scaling but not volume. Answer C appears to multiply the area scaling factor by the linear factor (

2.25 \times 2

), which has no geometric basis. Remember: linear dimensions scale by the factor, areas by the factor squared, and volumes by the factor cubed. Always identify which type of measurement you're dealing with first.

Question 15

A cone has its radius doubled and height tripled. If the original cone could hold 50 mL of liquid, how much can the new cone hold?

300 mL
600 mL (correct answer)
900 mL
1200 mL

Explanation: Volume of a cone is V = (1/3)πr²h. When radius doubles and height triples: new volume = (1/3)π(2r)²(3h) = (1/3)π(4r²)(3h) = 12 × (1/3)πr²h = 12 times original volume. Therefore: 12 × 50 = 600 mL. Choice A uses 6 times (2×3) instead of 12. Choice C uses 18 times, perhaps from 2²×3 + 2×3. Choice D uses 24 times, possibly from 2³×3.

Question 16

A cylindrical container is redesigned so that its radius decreases by 10% and its height increases by 32%. What is the approximate percentage change in volume?

Volume increases by 7% (correct answer)
Volume increases by 22%
Volume decreases by 3%
Volume decreases by 7%

Explanation: New radius = 0.9r, new height = 1.32h. New volume = π(0.9r)²(1.32h) = π(0.81r²)(1.32h) = 1.0692πr²h ≈ 1.07 times original volume. This represents about a 7% increase. Choice B might come from adding 32% - 10% = 22%. Choice C and D represent common errors in combining the percentage changes without proper consideration of the quadratic effect of radius.

Question 17

A pyramid with a square base has its base edge length tripled and its height reduced by half. How does the new volume compare to the original volume?

The new volume is 1.5 times the original
The new volume is 3 times the original
The new volume is 4.5 times the original (correct answer)
The new volume is 6 times the original

Explanation: Volume of a square pyramid is V = (1/3) × base area × height = (1/3) × s² × h. When base edge is tripled, base area becomes (3s)² = 9s². When height is halved, new height is h/2. New volume = (1/3) × 9s² × (h/2) = (9/2) × (1/3)s²h = 4.5 times the original volume. Choice A incorrectly adds the scaling effects. Choice B uses only the linear scaling factor. Choice D multiplies 3 × 2 instead of considering the area and height effects properly.

Question 18

A sphere is inscribed in a cube so that it touches all six faces. If the cube's edge length is doubled, by what factor does the volume of the inscribed sphere increase?

$2$
$4$
$6$
$8$ (correct answer)

Explanation: When a sphere is inscribed in a cube, the sphere's diameter equals the cube's edge length. If the cube's edge doubles, the sphere's diameter (and radius) also doubles. Since volume scales as the cube of linear dimensions, the sphere's volume increases by a factor of 2³ = 8. Choice A uses linear scaling. Choice B uses area scaling. Choice C incorrectly relates to the six faces.

Question 19

A cube with side length $s$ has its volume tripled by increasing each dimension by the same amount $x$ . Which equation represents this transformation?

$3(s + x) = s^3$
$(s + x)^3 = s^3 + 3x^3$
$(s + x)^3 = 3s^3$ (correct answer)
$s + 3x = 3s$

Explanation: This problem tests your understanding of volume scaling and algebraic equation setup. When dealing with three-dimensional transformations, you need to carefully track how changes in linear dimensions affect volume. Start by identifying what you know: the original cube has side length

s

and volume

s^3

. After increasing each dimension by

x

, the new cube has side length

(s + x)

and volume

(s + x)^3

. The problem states this new volume equals three times the original volume, so you need the equation

(s + x)^3 = 3s^3

. Choice C correctly represents this relationship. The left side gives the new volume after the transformation, and the right side gives three times the original volume. Choice A incorrectly treats the problem as if you're working with linear dimensions rather than volume. It multiplies the new side length by 3 instead of cubing it, completely missing the three-dimensional nature of volume. Choice B represents a common algebraic mistake. This equation suggests the new volume equals the original volume plus

3x^3

, which would only be true if volume increased additively rather than according to the cubic relationship

(s + x)^3

. Choice D treats this as a linear problem about side lengths rather than volumes. It ignores the fact that when you triple a cube's volume, you're not simply adding dimensions linearly. Remember: volume problems involving cubes require you to cube the side length. When you see "volume tripled," set up your equation so the new volume expression equals 3 times the original volume expression.