PSAT Math Multiple-Choice Question 88: Answer and Explanation

Question: 88

Three spherical balls with radius r are contained in a rectangular box. Two of the balls are each touching 5 sides of the rectangular box and the middle ball. The middle ball also touches four sides of the rectangular box. What is the volume of the space between the balls and the rectangular box?

(Note: The volume of a rectangular solid is given by the equation V = lwh. The volume of a sphere is given by the equation V = πr³).

• A. r³(3 - 4π)
• B. 4r²(14 - π)
• C. 4r³(6 - π)
• D. 12r²(r - π)

Correct Answer: C

Explanation:

C

The question asks for the volume of the space between the spheres and the rectangular box. Since no figure is provided, start by drawing the figure according to the description in the question. The figure should look like the following:

There are variables in the answer choices, so plug in. Make r = 2. Therefore, the diameter of the sphere is 4, which is also the width and height of the box. There are 3 spheres in a row, so the length of the box is 3(4) = 12. The formula for volume of a rectangular solid is V =lwh, so plug in the values for the length, width, and height to get (12)(4)(4) = 192. The formula for volume of a sphere is . So, the volume of each sphere is . There are 3 spheres, so multiply this result by 3 to get a total volume of 32π. Therefore, the volume of the spaces between the balls and the box is 192 - 32π. This is the target value; circle it. Now plug r = 2 into the answer choices to see which one matches the target value. Choice (A) becomes (2)³(3 - 4π) = 8(3 - 4π). Distribute the 8 to get 24 - 32π. This does not match the target, so eliminate (A). Choice (B) becomes 4(2)²(14 - π) = 16(14 - π). Distribute the 16 to get 224 - 16π. Eliminate (B). Choice (C) becomes 4(2)³ (6 - π) = 32(6 - π). Distribute the 32 to get 192 - 32π. Keep (C), but check (D) just in case. Choice (D) becomes 12(2)² (2 - π) = 48(2 - π). Distribute the 48 to get 96 - 48π. Eliminate (D). The correct answer is (C).