23. The figure shows a funnel which consists of a right circular cylinder and an inverted right circular cone with the same base diameter. The height and the base radius of the circular cylinder are 4 cm and 8 cm respectively while the slant height of the circular cone is 17 cm . (a) Find the curved surface area of the funnel. (b) The funnel is full of water initially. Then the water runs out at a rate of \( \pi \mathrm{cm}^{3} \) per minute. How long does it take to empty the funnel?

To tackle your questions about the funnel, let’s break it down into two parts! The curved surface area of the funnel can be calculated by finding the curved surfaces of both the cylinder and the cone. For the cylinder, the curved surface area (CSA) is given by \( CSA_{cylinder} = 2\pi rh \), where \( r = 8 \) cm and \( h = 4 \) cm. Thus, \( CSA_{cylinder} = 2\pi(8)(4) = 64\pi \) cm². For the cone, the curved surface area is \( CSA_{cone} = \pi r l \), where \( l = 17 \) cm. Therefore, \( CSA_{cone} = \pi(8)(17) = 136\pi \) cm². Adding both gives the total curved surface area of the funnel: \( CSA_{funnel} = 64\pi + 136\pi = 200\pi \) cm². Next, let’s determine how long it takes to empty the funnel. The volume of the funnel consists of the volume of both the cylinder and the cone. The volume of the cylinder is \( V_{cylinder} = \pi r^2 h = \pi (8)^2 (4) = 256\pi \) cm³. The volume of the cone is \( V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (8)^2 (17) = \frac{1368\pi}{3} = 456\pi \) cm³. Therefore, the total volume of the funnel is \( V_{funnel} = 256\pi + 456\pi = 712\pi \) cm³. If the water runs out at a rate of \( \pi \) cm³ per minute, then the total time to empty the funnel is \( \frac{712\pi}{\pi} = 712 \) minutes. In conclusion: (a) The curved surface area of the funnel is \( 200\pi \) cm². (b) It takes 712 minutes to empty the funnel.