((d)) A cylinder has a height of 8 cm and a redius of 7 cm . The height remains constant but the radius is doubled. (1) What is the volume of the enlarged cylinder? (2) How does the volume of the larger cylinder relate to tho volume of the origina! cylinder? (3) What is the surface area of the enlarged cylinder? (4) How does the surface area of the larger cyllinder relate to the surface area of the original cylinder?

To solve the problems regarding the cylinder, we will use the formulas for the volume and surface area of a cylinder. ### Given: - Original height \( h = 8 \) cm - Original radius \( r = 7 \) cm ### Formulas: 1. **Volume of a cylinder**: \[ V = \pi r^2 h \] 2. **Surface area of a cylinder**: \[ SA = 2\pi r(h + r) \] ### (1) Volume of the enlarged cylinder: When the radius is doubled, the new radius \( r' = 2r = 2 \times 7 = 14 \) cm. The height remains the same at \( h = 8 \) cm. Now, we calculate the volume of the enlarged cylinder: \[ V' = \pi (r')^2 h = \pi (14)^2 (8) = \pi (196)(8) = 1568\pi \text{ cm}^3 \] ### (2) Relation of the volume of the larger cylinder to the original cylinder: First, we calculate the volume of the original cylinder: \[ V = \pi (r^2) h = \pi (7^2)(8) = \pi (49)(8) = 392\pi \text{ cm}^3 \] Now, we find the ratio of the volumes: \[ \text{Ratio} = \frac{V'}{V} = \frac{1568\pi}{392\pi} = \frac{1568}{392} = 4 \] Thus, the volume of the enlarged cylinder is **4 times** the volume of the original cylinder. ### (3) Surface area of the enlarged cylinder: Using the new radius \( r' = 14 \) cm and height \( h = 8 \) cm, we calculate the surface area of the enlarged cylinder: \[ SA' = 2\pi r'(h + r') = 2\pi (14)(8 + 14) = 2\pi (14)(22) = 2\pi (308) = 616\pi \text{ cm}^2 \] ### (4) Relation of the surface area of the larger cylinder to the original cylinder: Now, we calculate the surface area of the original cylinder: \[ SA = 2\pi r(h + r) = 2\pi (7)(8 + 7) = 2\pi (7)(15) = 2\pi (105) = 210\pi \text{ cm}^2 \] Now, we find the ratio of the surface areas: \[ \text{Ratio} = \frac{SA'}{SA} = \frac{616\pi}{210\pi} = \frac{616}{210} \approx 2.9333 \] Thus, the surface area of the enlarged cylinder is approximately **2.93 times** the surface area of the original cylinder. ### Summary: 1. Volume of the enlarged cylinder: \( 1568\pi \) cm³ 2. Volume relation: Enlarged cylinder volume is 4 times the original. 3. Surface area of the enlarged cylinder: \( 616\pi \) cm² 4. Surface area relation: Enlarged cylinder surface area is approximately 2.93 times the original.