Final answer:
To find the length of a base edge of the pyramid, we equate the surface area formulas of the cone and pyramid. This results in a quadratic equation that is solved to find the length. The length of the base edge is approximately 7.94 inches.
Step-by-step explanation:
To find the length of a base edge of the pyramid, we need to compare the surface areas of the cone and the pyramid. The formula for the surface area of a cone is:
A = πr² + πrl, where r is the radius and l is the slant height.
The formula for the surface area of a square pyramid is:
A = 2lw + l², where l is the length of the base edge and w is the slant height.
Since both solids have the same surface area, we can equate the two formulas:
πr² + πrl = 2lw + l².
Substituting the given values, we have:
π(4²) + π(4)(5) = 2l(5) + l².
Simplifying this equation, we get:
π(16) + π(20) = 10l + l².
Combining like terms, we have:
16π + 20π = 10l + l².
36π = 10l + l².
Rearranging the equation and setting it equal to zero, we have:
l² + 10l - 36π = 0.
Using the quadratic formula, we can solve for l:
l = (-b ± √(b² - 4ac)) / (2a), where a = 1, b = 10, and c = -36π.
Calculating the discriminant, we have:
b² - 4ac = 100 - 4(-36π) = 400 + 144π.
Taking the square root, we get:
√(400 + 144π) ≈ 24.87.
Substituting this value into the quadratic formula, we have:
l ≈ (-10 ± 24.87) / 2.
Taking the positive root, the length of the base edge of the pyramid is approximately:
l ≈ (24.87 - 10) / 2 ≈ 7.94 inches.