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 find the surface area covered with felt on the hat, we must consider both the lateral surface area of the cylinder and the surface area of the brim. 1. **Lateral Surface Area of the Cylinder**: The formula for the lateral surface area of a cylinder is given by \( A = 2\pi rh \), where \( r \) is the radius and \( h \) is the height. - Here, \( r = 11 \) cm and \( h = 19 \) cm. - So, \( A_{cylinder} = 2\pi(11)(19) \). 2. **Area of the Top of the Cylinder**: The top of the cylinder is a circle, calculated using \( A = \pi r^2 \): - \( A_{top} = \pi(11)^2 \). 3. **Surface Area of the Brim**: The brim is essentially a ring, so we can find its area by calculating the area of the outer circle (including the brim) minus the area of the inner circle (the base of the cylinder): - The outer radius with the brim is \( 11 + 7 = 18 \) cm. - The area of the outer circle: \( A_{outer} = \pi(18)^2 \). - The area of the inner circle: \( A_{inner} = \pi(11)^2 \). - So, the area of the brim: \( A_{brim} = A_{outer} - A_{inner} = \pi(18^2 - 11^2) \). Now, let's compute the total area covered with felt: \[ A_{total} = A_{cylinder} + A_{top} + A_{brim} \] Putting it all together: 1. Calculate: - \( A_{cylinder} = 2\pi(11)(19) = 418\pi \) cm². - \( A_{top} = \pi(11)^2 = 121\pi \) cm². - \( A_{brim} = \pi(18^2 - 11^2) = \pi(324 - 121) = 203\pi \) cm². Combining these: \[ A_{total} = 418\pi + 121\pi + 203\pi = 742\pi \] So the final surface area covered with felt is: \[ A_{total} \approx 7388.56 \text{ cm}² \quad (\text{using } \pi \approx 3.14) \] Thus, the surface area covered with felt on the hat is approximately **7388.56 cm²**.