$40$ . DIG DEEPER A sphere with a radius of $2$ inches is inscribed in a right cone with a height of $6$ inches. Find the surface area and the volume of the cone.

Full solution

40

. DIG DEEPER A sphere with a radius of

2

inches is inscribed in a right cone with a height of

6

inches. Find the surface area and the volume of the cone.

Find Slant Height: First, let's find the slant height of the cone using the Pythagorean theorem. The radius of the sphere is the same as the radius of the cone's base. $\newline$ Slant height $l$ = $\sqrt{\text{radius}^2 + \text{height}^2}$ = $\sqrt{2^2 + 6^2}$ = $\sqrt{4 + 36}$ = $\sqrt{40}$ .
Calculate Surface Area: Now, calculate the surface area of the cone (excluding the base). The formula for the lateral surface area of a cone is $\pi r l$ . $\newline$ Surface area = $\pi \times \text{radius} \times \text{slant height} = \pi \times 2 \times \sqrt{40}$ .
Simplify Surface Area: Simplify the surface area calculation. $\newline$ Surface area = $2\pi \cdot \sqrt{40} = 2\pi \cdot \sqrt{4 \cdot 10} = 2\pi \cdot 2 \cdot \sqrt{10} = 4\pi \cdot \sqrt{10}$ .
Calculate Volume: Next, calculate the volume of the cone. The formula for the volume of a cone is $(\frac{1}{3})\pi r^2h$ . $\text{Volume} = (\frac{1}{3})\pi \times \text{radius}^2 \times \text{height} = (\frac{1}{3})\pi \times 2^2 \times 6 = (\frac{1}{3})\pi \times 4 \times 6 = 8\pi$ .
Check Answer: Finally, let's check if we've answered the question prompt correctly. $\newline$ We found the surface area (excluding the base) and the volume of the cone.