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ACT Math Help: Volume And Surface Area

Review real example questions for Volume And Surface Area in ACT Math.

Question 1 / 10

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A rectangular fish tank has a base measuring 12 inches by 10 inches. When a rock is fully submerged, the water level rises exactly 2 inches. What is the volume of the rock, in cubic inches?

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Question 1

A rectangular fish tank has a base measuring 12 inches by 10 inches. When a rock is fully submerged, the water level rises exactly 2 inches. What is the volume of the rock, in cubic inches?

  1. 24
  2. 120
  3. 240 (correct answer)
  4. 480

Explanation: This is a volume and displacement question testing Archimedes' principle (water displacement). Choice C (240) is correct — the volume of the submerged rock equals the volume of water displaced. The water level rose 2 inches across a 12 × 10 inch base. Volume displaced = 12 × 10 × 2 = 240 cubic inches = volume of rock. Choice A (24) adds the dimensions instead of multiplying: 12 + 10 + 2 = 24. Choice B (120) uses only the base area without the height rise: 12 × 10 = 120 — finding the base area but not the volume of displaced water. Choice D (480) doubles the correct answer — perhaps computing 12 × 10 × 4 (using 4 instead of 2) or multiplying the result by 2. Pro tip: When an object is submerged in a tank, the volume of the displaced water equals the volume of the object. Displaced water forms a rectangular prism with the tank's base dimensions and the height equal to the water rise. Volume = length × width × rise = 12 × 10 × 2 = 240. This principle applies whenever the tank has uniform (rectangular) cross-section.

Question 2

A rectangular prism has length 9 units, width 4 units, and height 2 units. What is the surface area of the rectangular prism?​

  1. 72 square units
  2. 124 square units (correct answer)
  3. 152 square units
  4. 104 square units

Explanation: We need to find the surface area of a rectangular prism with length 9, width 4, and height 2 units. The surface area formula is SA = 2(lw + lh + wh). Substituting: SA = 2(9×4 + 9×2 + 4×2) = 2(36 + 18 + 8) = 2(62) = 124 square units. Choice A incorrectly calculated only the sum of areas without doubling for opposite faces.

Question 3

A right circular cylinder has a volume of 72π72\pi cubic inches and a height of 8 inches. What is the radius, in inches, of the cylinder's base?

  1. 3 (correct answer)
  2. 4.5
  3. 6
  4. 9

Explanation: This is a cylinder volume question testing inverse use of the volume formula. Choice A (3) is correct — V = πr²h → 72π = πr²(8) → divide both sides by π: 72 = 8r² → r² = 9 → r = 3 inches. Choice B (4.5) results from dividing 72 by 8 to get 9, then halving instead of taking the square root: 9/2 = 4.5. Choice C (6) may result from computing 72/8 = 9 and then computing √(9 × 4) = 6 — an incorrect extra step. Choice D (9) correctly solves r² = 9 but reports r² rather than r — forgetting to take the square root. Pro tip: When solving V = πr²h for r, cancel π first (it divides out cleanly), then divide by h to isolate r², and THEN take the square root. Write out each step to avoid stopping at r².

Question 4

A rectangular prism has length 12 units, width 5 units, and height 3 units. All measurements are in units. What is the volume of the rectangular prism?

  1. 60 cubic units
  2. 180 cubic units (correct answer)
  3. 120 cubic units
  4. 360 cubic units

Explanation: The solid is a rectangular prism with length 12 units, width 5 units, and height 3 units, and we need to find its volume. The formula for the volume of a rectangular prism is V = lwh, where l is length, w is width, and h is height. Substituting the values, V = 12·5·3 = 60·3. This calculates to 180 cubic units. Confusing with surface area might lead to other values like 2(lw + lh + wh) = 222.

Question 5

A cone has radius 3 units and height 12 units. What is the volume?

  1. 108π108\pi cubic units
  2. 36π36\pi cubic units (correct answer)
  3. 45π45\pi cubic units
  4. 108 cubic units

Explanation: This problem asks for the volume of a cone with radius 3 units and height 12 units. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. Substituting the given values: V = (1/3)π(3²)(12) = (1/3)π(9)(12) = (1/3)π(108) = 36π cubic units. Choice A might result from forgetting the 1/3 factor in the cone volume formula.

Question 6

A sphere has radius 3 units. What is the volume of the sphere? (Use π\pi in your answer.)

  1. 36π cubic units36\pi\text{ cubic units} (correct answer)
  2. 27π cubic units27\pi\text{ cubic units}
  3. 12π cubic units12\pi\text{ cubic units}
  4. 43π cubic units\frac{4}{3}\pi\text{ cubic units}

Explanation: We need to find the volume of a sphere with radius 3 units. The volume formula for a sphere is V = (4/3)πr³. Substituting r = 3: V = (4/3)π(3³) = (4/3)π(27) = (4 × 27/3)π = (108/3)π = 36π cubic units.

Question 7

A rectangular prism has length 6 units, width 4 units, and height 3 units. What is the volume?

  1. 72 cubic units (correct answer)
  2. 52 cubic units
  3. 84 cubic units
  4. 96 cubic units

Explanation: This is finding the volume of a rectangular prism with length 6, width 4, and height 3 units. The volume formula for a rectangular prism is V = lwh, where l is length, w is width, and h is height. Substituting the values: V = 6 × 4 × 3 = 72 cubic units. This involves multiplying all three dimensions together.

Question 8

A square pyramid has a square base with side length 6 units and a vertical height of 9 units from the base to the apex. What is the volume of the pyramid in cubic units?

  1. 108 cubic units (correct answer)
  2. 324 cubic units
  3. 216 cubic units
  4. 72 cubic units

Explanation: The solid is a square pyramid with base side length 6 units and height 9 units, and we need to find its volume. The formula for the volume of a pyramid is V = (1/3)Bh, where B is the base area and h is the height. The base area B = 6² = 36, so V = (1/3)(36)(9) = (1/3)(324). This calculates to 108 cubic units. Forgetting the 1/3 factor might yield 324.

Question 9

A rectangular prism has dimensions 8 units, 6 units, and 5 units. What is the volume of the prism?

  1. 240 cubic units (correct answer)
  2. 96 cubic units
  3. 280 cubic units
  4. 120 cubic units

Explanation: We need to find the volume of a rectangular prism with dimensions 8, 6, and 5 units. The volume formula for a rectangular prism is V = lwh where l, w, and h are the dimensions. Substituting: V = 8 × 6 × 5 = 240 cubic units. This represents the space occupied by the three-dimensional object.

Question 10

What is the surface area of a cube with side length 10 units?

  1. 300 square units
  2. 1000 square units
  3. 600 square units (correct answer)
  4. 400 cubic units

Explanation: We need to find the surface area of a cube with side length 10 units. The surface area formula for a cube is SA = 6s² where s is the side length. Substituting s = 10: SA = 6(10²) = 6(100) = 600 square units. A cube has six square faces, each with area s².