For a right circular cylinder, the ratio of curved surface area to volume is 1 : 7. The ratio of total surface area to volume is 187 : 770. What is the ratio of the base radius to the height of the cylinder?

Introduction / Context:
This question links the curved surface area and total surface area of a right circular cylinder with its volume using ratios. The goal is to determine the ratio of the base radius to the height. It is a good test of how well one can convert verbal ratio conditions into algebraic equations and solve them systematically using standard cylinder formulas.

Given Data / Assumptions:

• Right circular cylinder with radius r and height h.
• Curved surface area (CSA) to volume (V) ratio is 1 : 7.
• Total surface area (TSA) to volume ratio is 187 : 770.
• We need the ratio r : h.

Concept / Approach:
For a cylinder, curved surface area C = 2 * pi * r * h, volume V = pi * r^2 * h, and total surface area T = C + 2 * pi * r^2 = 2 * pi * r * h + 2 * pi * r^2. The ratio C : V can be simplified to obtain a relation between r and h. Then the ratio T : V gives another equation, which helps in finding the exact numeric ratio between r and h. The presence of nice rational ratios suggests that r and h will have a simple integer ratio solution.

Step-by-Step Solution:
Step 1: Curved surface area C = 2 * pi * r * h and volume V = pi * r^2 * h.Step 2: Given C : V = 1 : 7, so C / V = 1 / 7.Step 3: Compute C / V = (2 * pi * r * h) / (pi * r^2 * h) = 2 / r. So 2 / r = 1 / 7, which gives r = 14.Step 4: Total surface area T = 2 * pi * r * h + 2 * pi * r^2. Thus T / V = (2 * pi * r * h + 2 * pi * r^2) / (pi * r^2 * h) = (2 / r) + (2 / h).Step 5: We already know 2 / r = 1 / 7 from the first ratio, so 2 / r = 1 / 7.Step 6: Given T : V = 187 : 770, so T / V = 187 / 770. Simplify 187 / 770 to 17 / 70.Step 7: Use T / V = (2 / r) + (2 / h) = 1 / 7 + 2 / h = 17 / 70.Step 8: Convert 1 / 7 to denominator 70: 1 / 7 = 10 / 70. So 10 / 70 + 2 / h = 17 / 70 gives 2 / h = 7 / 70 = 1 / 10.Step 9: Therefore h = 10. Since r = 14, ratio r : h = 14 : 20 = 7 : 10.

Verification / Alternative check:
Using r = 14 and h = 20, C = 2 * pi * 14 * 20 = 560pi and V = pi * 14^2 * 20 = 3920pi. Then C : V = 560pi : 3920pi = 1 : 7, which matches the first ratio. T = C + 2 * pi * r^2 = 560pi + 2 * pi * 196 = 560pi + 392pi = 952pi. Then T : V = 952pi : 3920pi, which simplifies to 952 : 3920 = 17 : 70 = 187 : 770, as required. Both ratios are satisfied, confirming r : h = 7 : 10.

Why Other Options Are Wrong:

• 5 : 8, 4 : 9, and 3 : 7 do not satisfy both ratio conditions when substituted into the formulas for C and T.
• 2 : 5 leads to values of r and h that fail to produce the required fractions 1/7 and 187/770 for C/V and T/V.

Common Pitfalls:

• Using an incorrect expression for T / V and forgetting to include both curved surface area and base areas.
• Not simplifying the ratio 187 : 770 correctly to 17 : 70, which is essential for clean algebra.
• Attempting to guess r : h without verifying both given ratios.

Final Answer:
The ratio of the base radius to the height of the cylinder is 7 : 10.