When Rounding Changes the Answer Choice
A Math item asks: "A cylindrical water tank has an interior radius of 3.5 feet and a height of 12 feet. A pump fills the tank at a constant rate of 22 cubic feet per minute. To the nearest whole minute, how long does it take to fill the tank completely? Use π ≈ 3.14."
A student computes V = πr²h = 3.14 × 12.25 × 12. They get 12.25 × 12 = 147, then 147 × 3.14 = 461.58 cubic feet. Dividing: 461.58 ÷ 22 = 20.98 minutes, which rounds to 21 minutes.
The answer choices are: F) 19, G) 21, H) 22, J) 24, K) 27. The student picks G. A friend used π ≈ 3.14159 and got 461.81 ÷ 22 = 20.99, also rounding to 21. Both feel confident.
1. Verify the student's arithmetic and confirm whether G is defensible. Show the computation explicitly.
Read the worked answer
Check each step:
- r² = 3.5² = 12.25 ✓
- r²h = 12.25 × 12 = 147 ✓
- V = 3.14 × 147 = 461.58 ✓ (3 × 147 = 441; 0.14 × 147 = 20.58; sum = 461.58)
- Time = 461.58 ÷ 22. Compute: 22 × 20 = 440; remainder 21.58; 21.58 ÷ 22 ≈ 0.981; total ≈ 20.98 ✓
Rounding 20.98 to the nearest whole minute gives 21. Choice G is correct. The friend's more precise π yields 20.99, still 21. Both paths converge, so the answer is robust to the π approximation - a good sign the item was designed so the approximation doesn't flip the choice.
Note the trap: 20.98 is close to 21 but a careless student might round down to 20 (not offered) or mis-multiply and land on 22 (choice H) by rounding up twice. The distractors 19, 22, 24, 27 correspond to plausible errors (using diameter instead of radius gives r = 7, V ≈ 1846, time ≈ 84 - not matching, so 19/22 likely come from other slips like forgetting to square or using h as diameter).
2. Suppose a student instead used r = 7 (treating 3.5 as the diameter). What answer would they get, and which distractor would that likely produce? Explain the conceptual error.
Read the worked answer
If r = 7: r² = 49; × h = 49 × 12 = 588; × 3.14 = 1846.32 ft³. Time = 1846.32 ÷ 22 ≈ 83.9 → 84 minutes. That isn't among the choices, so a student making this error would be confused and likely re-guess.
The conceptual error: 3.5 ft is already the radius ("interior radius of 3.5 feet"). Doubling it to 7 wrongly treats the given radius as a diameter. Because volume scales with r², this quadruples the volume (factor of 4: 461.58 × 4 = 1846.3 ✓), so the time quadruples too (21 × 4 = 84).
The takeaway for ACT Math: read whether a dimension is radius or diameter before substituting, and sanity-check by asking whether your answer scales sensibly. Distractors are often built from exactly these radius/diameter and square/not-square slips, so matching a choice is not proof of correctness - the setup must justify it.