A conical structure and a hemispherical structure share the same base radius r. If the curved surface area of the cone is equal to the curved surface area of the hemisphere, express the slant height l of the cone in terms of r.

Correct Answer: l = 2r

Step-by-step explanation:

Let r be the common base radius of the conical structure and the hemispherical structure, and let l be the slant height of the cone.

The curved surface area (CSA) of a cone is calculated using the formula:

${\text{CSA}}_{\text{cone}}=\pi rl$

The curved surface area (CSA) of a hemisphere is calculated using the formula:

${\text{CSA}}_{\text{hemisphere}}=2\pi r^2$

According to the problem statement, the curved surface area of the cone is equal to the curved surface area of the hemisphere:

${\text{CSA}}_{\text{cone}}={\text{CSA}}_{\text{hemisphere}}$

Substituting the respective formulas into the equation:

$\pi rl=2\pi r^2$

To find the slant height l in terms of r, divide both sides of the equation by πr (since radius r > 0 and π ≠ 0):

$l=\frac{2\pi r^2}{\pi r}$

Simplifying the right-hand side:

$l=2r$

Therefore, the slant height l of the cone expressed in terms of r is l = 2r.