Geometry and Trigonometry practice question

Answer: A. 96π

The water forms a cone similar to the whole cup, so its radius scales with its height: r/9 = 8/12, giving r = 6. Its volume is (1/3)π(6²)(8) = (1/3)π(288) = 96π. (B) 144π = (1/3)π(6²)(12) — the student correctly scaled the radius to 6 but then used the full height 12 instead of the water depth 8. (C) 216π = (1/3)π(9²)(8) — the student used the depth 8 but kept the top radius 9, forgetting that the water surface is narrower; this is also the result of scaling the full volume 324π linearly by 8/12 instead of by (8/12)³. (D) 324π = (1/3)π(9²)(12) is the volume of the entire cup, i.e., ignoring the depth condition.

Why the other answers are wrong

B. 144π

You correctly scaled the water's radius down to 6, but then plugged in the cup's full height of 12 instead of the 8-inch water depth: (1/3)π(6²)(12) = 144π.

C. 216π

You used the water depth of 8 but kept the radius at 9 — the water surface is narrower than the cup's rim, so its radius must scale to 6. (This is also what you get by scaling the full volume by 8/12 instead of (8/12)³.)

D. 324π

That's (1/3)π(9²)(12), the volume of the entire cup — it ignores the fact that the water is only 8 inches deep.

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