Final answer:
To find the slant height of a cone whose height is equal to its base diameter, we use the Pythagorean theorem. The slant height (l) is found to be (√5)/2 times the height of the cone.
Step-by-step explanation:
The question asks to find the slant height of a cone whose height is equal to its base diameter. According to Pythagorean theorem in a right-angled triangle, the square of the slant height (l) is equal to the sum of the squares of the height (h) and the radius (r). Since the height is equal to the diameter, we can represent the radius (which is half the diameter) as h/2. Therefore:
l² = h² + (h/2)²
l² = h² + h²/4
l² = (5/4)h²
l = h * √(5/4)
l = h * (√5)/2
So, the slant height of the cone is (√5)/2 times the height of the cone.