Question 2 (2 points) Saved A hat is a cylinder with radius 11 cm and height 19 cm . A brim 7 cm wide is attached around the bottom of the cylinder. Felt is attached to the outside of the hat and the top of the brim. Determine the surface area covered with felt.

To calculate the surface area covered with felt, we first need to consider the shape of the hat. The hat is a cylinder with a radius (r) of 11 cm and a height (h) of 19 cm. The brim adds an additional 7 cm in width all around, thus effectively increasing the radius for the brim to 11 cm + 7 cm = 18 cm. The surface area of the cylinder (excluding the bottom) consists of two parts: the lateral surface area of the cylinder and the area of the top of the cylinder. The formula for the lateral surface area of a cylinder is given by \( 2 \pi r h \). 1. **Lateral Surface Area of the Hat:** - Radius = 11 cm - Height = 19 cm - Lateral surface area \( = 2 \pi (11)(19) \approx 2 \pi (209) \approx 1317.33 \, \text{cm}^2 \) 2. **Area of the Top of the Cylinder:** - Area \( = \pi r^2 = \pi (11^2) = \pi (121) \approx 380.13 \, \text{cm}^2 \) 3. **Surface Area of the Brim:** - The brim can be viewed as a circular band. To calculate its lateral surface area, we consider it as a cylinder with: * An outer radius of 18 cm and an inner radius of 11 cm, and a height of 7 cm (the width of the brim). - We calculate the lateral surface area for this outer part and subtract the inner part: \[ \text{Lateral surface area of outer cylinder} = 2\pi(18)(7) \approx 2\pi(126) \approx 792.42 \, \text{cm}^2 \] \[ \text{Lateral surface area of inner cylinder} = 2\pi(11)(7) \approx 2\pi(77) \approx 483.02 \, \text{cm}^2 \] \[ \text{Brim surface area} = 792.42 - 483.02 \approx 309.40 \, \text{cm}^2 \] 4. **Total Surface Area Covered with Felt:** \[ \text{Total} = \text{Lateral Surface Area of Hat} + \text{Top Area of Hat} + \text{Brim Surface Area} \] \[ \text{Total} \approx 1317.33 + 380.13 + 309.40 \approx 2006.86 \, \text{cm}^2 \] Thus, the surface area covered with felt is approximately **2006.86 cm².**