Difficulty: Hard
Correct Answer: 4312*sqrt(6)
Explanation:
Introduction / Context:This question combines cylinder surface-area formulas with ratio reasoning. The ratio CSA:TSA = 2:5 immediately gives a relationship between height and radius, and then the given TSA lets us compute the exact radius. Finally, volume follows from V = pi*r^2*h. This is a classic multi-step mensuration problem.
Given Data / Assumptions:
Concept / Approach:Use the ratio to connect CSA and TSA, then derive h in terms of r using TSA/CSA = (h + r)/h. Use CSA to find r, then compute volume.
Step-by-Step Solution:
Step 1: Since CSA:TSA = 2:5 and TSA = 3080, CSA = (2/5)*3080 = 1232. Step 2: TSA/CSA = 3080/1232 = 5/2. Step 3: TSA/CSA = (2*pi*r*(h + r)) / (2*pi*r*h) = (h + r)/h. Step 4: So (h + r)/h = 5/2 => 1 + (r/h) = 5/2 => r/h = 3/2 => h = (2/3)*r. Step 5: Use CSA = 2*pi*r*h = 2*pi*r*(2r/3) = (4/3)*pi*r^2 = 1232. Step 6: r^2 = 1232*3/(4*pi) = 924/pi. With pi = 22/7, r^2 = 924*7/22 = 294. Step 7: r = sqrt(294) = 7*sqrt(6). Then h = (2/3)*r = (14/3)*sqrt(6). Step 8: Volume V = pi*r^2*h = pi*294*(14/3)*sqrt(6) = pi*1372*sqrt(6). Step 9: With pi = 22/7, V = 1372*(22/7)*sqrt(6) = 4312*sqrt(6).Verification / Alternative check:Check TSA using r = 7*sqrt(6) and h = (2/3)r: TSA becomes 3080 cm^2 exactly (with pi = 22/7), confirming consistency.
Why Other Options Are Wrong:
Other coefficients (4522, 3822, 4642, 4112): come from wrong CSA value, wrong ratio inversion, or wrong pi usage.Common Pitfalls:Mixing CSA with TSA, forgetting the 2*pi*r factor cancels in the ratio, or using (2/5) incorrectly as (5/2).
Final Answer:4312*sqrt(6)
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