1. **State the problem:** We know a can of beans has a circular base with area 167 square inches and a height of 8 inches. We want to find the area of the label that covers the entire side of the can, which is the lateral surface area of the cylinder. 2. **Recall the formulas:** - Area of base circle: $A = \pi r^2$ - Lateral surface area of cylinder: $L = 2 \pi r h$ 3. **Find the radius $r$ from the base area:** Given $A = 167$, $$167 = \pi r^2 \implies r^2 = \frac{167}{\pi}$$ 4. **Calculate the lateral surface area:** Using $L = 2 \pi r h$, substitute $r = \sqrt{\frac{167}{\pi}}$ and $h = 8$: $$L = 2 \pi \times \sqrt{\frac{167}{\pi}} \times 8 = 16 \pi \sqrt{\frac{167}{\pi}}$$ Simplify the square root: $$\sqrt{\frac{167}{\pi}} = \frac{\sqrt{167}}{\sqrt{\pi}}$$ So, $$L = 16 \pi \times \frac{\sqrt{167}}{\sqrt{\pi}} = 16 \sqrt{167} \times \frac{\pi}{\sqrt{\pi}} = 16 \sqrt{167 \pi}$$ 5. **Interpret the expression:** $16 \sqrt{167 \pi}$ is the exact lateral surface area in square inches. 6. **Check available choices:** Choices given are $32\pi$, $64$, $64\pi$. Since these don't match the exact value, let's approximate to verify: - $r = \sqrt{167/\pi} \approx \sqrt{53.15} \approx 7.29$ inches - $L = 2 \pi r h = 2 \pi \times 7.29 \times 8 \approx 2 \pi \times 58.32 \approx 366.4$ square inches Given choices are significantly lower, none match the correct lateral surface area. 7. **Conclusion:** The lateral surface area of the can is approximately 366.4 square inches, so none of the provided multiple-choice answers are correct for the label area. **Final answer:** The area of the label covering the side is approximately $366.4$ square inches.