In a right circular cone, the radius is r and the height is h. They satisfy the relation 4r = h + √(r^2 + h^2), where √(r^2 + h^2) represents the slant height. If r is not zero, what is the ratio r : h?

Introduction / Context:
This algebraic geometry problem comes from the context of a right circular cone, where the radius r, height h, and slant height l are related by l = √(r^2 + h^2). The question gives a special relation between r, h, and the slant height: 4r = h + √(r^2 + h^2). We are asked to find the ratio r : h that satisfies this relation, assuming r is not zero. This tests the ability to interpret a geometric formula, substitute the expression for slant height, and then solve a non linear equation by using a suitable substitution.

Given Data / Assumptions:
- Right circular cone with radius r and height h. - Slant height l is given by l = √(r^2 + h^2). - Given relation: 4r = h + √(r^2 + h^2). - r is not equal to zero and r, h are positive real numbers. - We must determine r : h.

Concept / Approach:
We can use the expression for slant height directly in the given relation. To simplify the equation and avoid dealing with both r and h separately, we introduce a ratio variable k = r / h. We then express r in terms of h and k and rewrite the relation. This turns the equation into one involving only k. After simplification, we solve for k and match it to one of the given ratio options. Finally, we interpret k as r : h.

Step-by-Step Solution:
Step 1: Given relation is 4r = h + √(r^2 + h^2). Step 2: Introduce k = r / h. Then r = k * h. Step 3: Substitute r = k * h into the relation. We get 4 * (k * h) = h + √((k * h)^2 + h^2). Step 4: Simplify inside the square root: (k * h)^2 + h^2 = k^2 * h^2 + h^2 = h^2 * (k^2 + 1). Step 5: So √(r^2 + h^2) becomes √(h^2 * (k^2 + 1)) = h * √(k^2 + 1). Step 6: The equation now is 4k * h = h + h * √(k^2 + 1). Step 7: Since h is positive and not zero, divide both sides by h to get 4k = 1 + √(k^2 + 1). Step 8: Isolate the square root: √(k^2 + 1) = 4k − 1. Step 9: Square both sides to remove the square root: k^2 + 1 = (4k − 1)^2. Step 10: Expand the square: (4k − 1)^2 = 16k^2 − 8k + 1. Step 11: Set up the equation k^2 + 1 = 16k^2 − 8k + 1. Step 12: Subtract k^2 + 1 from both sides: 0 = 15k^2 − 8k. Step 13: Factor out k: 0 = k * (15k − 8). Step 14: Since r is not zero, k = r / h is not zero, so k cannot be 0. Hence we must have 15k − 8 = 0. Step 15: Solve 15k − 8 = 0 to get k = 8 / 15. Step 16: Therefore r / h = 8 / 15, so the ratio r : h is 8 : 15.

Verification / Alternative check:
To verify, take r : h = 8 : 15. Let r = 8 and h = 15 for simplicity. Then l = √(r^2 + h^2) = √(64 + 225) = √289 = 17. Check the relation: left side 4r = 4 * 8 = 32. Right side h + l = 15 + 17 = 32. Both sides match, confirming that the ratio 8 : 15 satisfies the given condition.

Why Other Options Are Wrong:
Option 17 : 8 would correspond to k = 17 / 8, which does not satisfy the equation 4k = 1 + √(k^2 + 1). Option 8 : 17 gives k = 8 / 17, which fails to satisfy the squared relation derived from the original equation. Option 15 : 8 gives k = 15 / 8. If we substitute this into 4k, we get 60 / 8 = 7.5, which does not equal 1 + √(k^2 + 1), so it is incorrect.

Common Pitfalls:
A frequent mistake is to forget that squaring an equation can introduce extraneous solutions, which is why we must check the final value of k in the original relation. Another possible error is incorrect algebra during expansion of (4k − 1)^2 or when rearranging terms. Introducing the ratio k = r / h early is a powerful technique for simplifying problems involving geometric relations between length quantities.

Final Answer:
The ratio of radius to height is r : h = 8 : 15.