A joker's cap is in the form of a right circular cone of base radius $7 \mathrm{~cm}$ and height $24 \mathrm{~cm}$. Find the area of the sheet required to make 10 such caps.

Step 1 ：Given a joker's cap in the form of a right circular cone with base radius \(r = 7 \mathrm{~cm}\) and height \(h = 24 \mathrm{~cm}\).

Step 2 ：Find the slant height \(l\) using the Pythagorean theorem: \(l = \sqrt{r^2 + h^2}\).

Step 3 ：\(l = \sqrt{7^2 + 24^2} = 25 \mathrm{~cm}\)

Step 4 ：Find the surface area of one cap using the formula: \(A = \pi r (r + l)\).

Step 5 ：\(A = \pi (7)(7 + 25) \approx 703.72 \mathrm{~cm^2}\)

Step 6 ：Find the area of the sheet required to make 10 caps by multiplying the surface area of one cap by 10.

Step 7 ：\(A_{10} = 10 \times 703.72 \approx 7037.17 \mathrm{~cm^2}\)

Step 8 ：\(\boxed{7037.17 \mathrm{~cm^2}}\) is the area of the sheet required to make 10 such caps.