If 2r = h + sqrt(r^2 + h^2) for a right circular cone, then what is the ratio r : h?

Introduction / Context:
This problem presents an algebraic relation between the radius r and height h of a right circular cone: 2r = h + sqrt(r^2 + h^2). The task is to find the ratio r : h that satisfies this equation. Even though the formula may look unusual at first glance, it is simply a nonlinear equation in terms of r and h. By converting to a single variable ratio, we can solve it systematically.

Given Data / Assumptions:
• Relation between radius r and height h: 2r = h + sqrt(r^2 + h^2). • We assume r > 0 and h > 0 since they represent geometric dimensions. • We must find the simplified ratio r : h.

Concept / Approach:
Because the equation involves r and h together with a square root, a good method is to express the relationship in terms of the ratio k = h / r or k = r / h. This reduces the equation to a single variable. Here we choose k = h / r. Then both h and sqrt(r^2 + h^2) can be expressed in terms of r and k. This allows us to simplify and solve for k using algebra. Once k is found, the ratio r : h can be deduced easily.

Step-by-Step Solution:
Step 1: Start from the given equation: 2r = h + sqrt(r^2 + h^2). Step 2: Introduce k = h / r, so h = k * r. Step 3: Substitute h = k * r into the equation: 2r = k * r + sqrt(r^2 + k^2 * r^2). Step 4: Factor r^2 inside the square root: sqrt(r^2 (1 + k^2)) = r * sqrt(1 + k^2). Step 5: The equation becomes 2r = k * r + r * sqrt(1 + k^2). Step 6: Divide both sides by r (nonzero): 2 = k + sqrt(1 + k^2). Step 7: Isolate the square root: sqrt(1 + k^2) = 2 - k. Step 8: Square both sides to eliminate the square root: 1 + k^2 = (2 - k)^2. Step 9: Expand the right side: (2 - k)^2 = 4 - 4k + k^2. Step 10: Equate: 1 + k^2 = 4 - 4k + k^2. Step 11: Subtract k^2 from both sides: 1 = 4 - 4k. Step 12: Rearrange: 4k = 4 - 1 = 3, so k = 3 / 4. Step 13: Recall k = h / r, so h / r = 3 / 4. Therefore r : h = 4 : 3.

Verification / Alternative Check:
Take r : h = 4 : 3. Let r = 4 and h = 3 for simplicity. Substitute into the left side: 2r = 2 * 4 = 8. Now evaluate the right side: h + sqrt(r^2 + h^2) = 3 + sqrt(4^2 + 3^2) = 3 + sqrt(16 + 9) = 3 + sqrt(25) = 3 + 5 = 8. Both sides are equal, confirming that the ratio r : h = 4 : 3 satisfies the original equation.

Why Other Options Are Wrong:
• 1 : 2, 2 : 3 and 3 : 5 do not satisfy the equation. Substituting these ratios into 2r = h + sqrt(r^2 + h^2) leads to unequal left and right sides. • Only r : h = 4 : 3 makes the equation hold exactly.

Common Pitfalls:
Common errors include squaring incorrectly, forgetting to divide by r, or solving for the wrong ratio (r / h instead of h / r) and then misinterpreting the result. It is also important after squaring to check that the solution is valid in the original equation, because squaring can sometimes introduce extraneous solutions. In this case, the solution k = 3 / 4 checks out perfectly.

Final Answer:
The required ratio of radius to height is r : h = 4 : 3.