22. Find the amount of paper needed for the label on the can. Use 3.14 for \( \pi \) and round to the nearest tenth. A \( 21.2 \mathrm{in}^{2} \) B \( 25.5 \mathrm{in}^{2} \) C \( 29.1 \mathrm{in}^{2} \) D \( 42.4 \mathrm{in}^{2} \) 23. A cylinder has a surface area of 5652 square millimeters. If the diameter of the cylinder is 30 millimeters, what is the height? Use 3.14 for \( \pi \). F 10 mm G 45 mm H 59 mm J 105 mm

To solve these problems, we will use the formulas for the surface area of a cylinder and the necessary calculations step by step. ### Problem 22: Find the amount of paper needed for the label on the can. The surface area \( A \) of a cylinder (which is the label area in this case) is given by the formula: \[ A = 2\pi r(h + r) \] Where: - \( r \) is the radius of the base of the cylinder. - \( h \) is the height of the cylinder. However, we need the dimensions of the cylinder to calculate the surface area. Since the problem does not provide these, we cannot solve it directly. ### Problem 23: Find the height of the cylinder. Given: - Surface area \( A = 5652 \, \text{mm}^2 \) - Diameter \( d = 30 \, \text{mm} \) First, we find the radius \( r \): \[ r = \frac{d}{2} = \frac{30}{2} = 15 \, \text{mm} \] Now, we can use the surface area formula for a cylinder: \[ A = 2\pi r(h + r) \] Substituting the known values: \[ 5652 = 2 \cdot 3.14 \cdot 15(h + 15) \] Now, we simplify and solve for \( h \): 1. Calculate \( 2 \cdot 3.14 \cdot 15 \): \[ 2 \cdot 3.14 \cdot 15 = 94.2 \] 2. Substitute back into the equation: \[ 5652 = 94.2(h + 15) \] 3. Divide both sides by 94.2: \[ h + 15 = \frac{5652}{94.2} \] 4. Calculate \( \frac{5652}{94.2} \): \[ h + 15 \approx 60 \] 5. Subtract 15 from both sides to find \( h \): \[ h \approx 60 - 15 = 45 \, \text{mm} \] Thus, the height of the cylinder is approximately \( 45 \, \text{mm} \). ### Final Answers: - For Problem 22, we cannot determine the answer without additional information. - For Problem 23, the height of the cylinder is \( 45 \, \text{mm} \), corresponding to option G. Let me know if you need further assistance!