The volumes of two right circular cones are in the ratio 1 : 4 and the diameters of their circular bases are in the ratio 4 : 5. What is the ratio of the heights of the two cones?

Introduction / Context:
This question checks your understanding of how the volume of a right circular cone depends on both its base radius and height. You are given the ratio of volumes and the ratio of diameters, and you must infer the resulting ratio of heights using proportional reasoning.

Given Data / Assumptions:

• Volume ratio of cones V1 : V2 = 1 : 4.
• Ratio of diameters d1 : d2 = 4 : 5.
• Therefore radii also satisfy r1 : r2 = 4 : 5.
• We need h1 : h2, the ratio of the heights.

Concept / Approach:
The volume of a cone is given by V = (1 / 3) * π * r^2 * h. For two cones, the ratio of their volumes equals the product of the ratio of the squares of their radii and the ratio of their heights. We set up an equation using the given ratios and solve for the height ratio h1 : h2.

Step-by-Step Solution:
Step 1: Write the volume ratio: V1 / V2 = (1 / 3 * π * r1^2 * h1) / (1 / 3 * π * r2^2 * h2) = (r1^2 * h1) / (r2^2 * h2).Step 2: Substitute the given volume ratio: (r1^2 * h1) / (r2^2 * h2) = 1 / 4.Step 3: Use r1 : r2 = 4 : 5, so r1^2 : r2^2 = 16 : 25.Step 4: Therefore (16 * h1) / (25 * h2) = 1 / 4.Step 5: Cross multiply to get 64 * h1 = 25 * h2.Step 6: Hence h1 / h2 = 25 / 64, so the ratio of heights is 25 : 64.

Verification / Alternative check:
You can assume a convenient value such as h1 = 25 and h2 = 64 and compute the corresponding volumes numerically to see that their ratio indeed becomes 1 : 4, confirming the derived relationship.

Why Other Options Are Wrong:
The ratios 1 : 5, 4 : 25 and 16 : 25 do not satisfy the volume ratio when combined with the squared radius ratio 16 : 25. Only the height ratio 25 : 64 produces the required overall volume ratio of 1 : 4.

Common Pitfalls:
A frequent mistake is to directly divide the volume ratio by the radius ratio instead of using the squared radius. Another error is to confuse diameter ratio with radius ratio. Remember that volume depends on r^2, not r alone.

Final Answer:
The heights of the two cones are in the ratio 25 : 64.