Let's go step by step as requested: --- ## **Step 1: Unit Conversion** **Given:** Capacity = 16 fluid ounces **Conversion:** 1 US fluid ounce = 1.80469 cubic inches So, \[ 16 \text{ fl oz} = 16 \times 1.80469 = 28.875 \text{ cubic inches} \] --- ## **Step 2: Volume Equation for Can** Let: - \( r \) = radius (in inches) - \( h \) = height (in inches) **Volume of cylinder:** \[ V = \pi r^2 h = 28.875 \] --- ## **Step 3: Cost Function** - **Side area:** \( 2\pi r h \) (cost: 4 cents/in²) - **Bottom area:** \( \pi r^2 \) (cost: 4 cents/in²) - **Top area:** \( \pi r^2 \) (cost: 2 cents/in²) **Total cost function:** \[ C = 4(2\pi r h) + 4(\pi r^2) + 2(\pi r^2) \] \[ C = 8\pi r h + 4\pi r^2 + 2\pi r^2 \] \[ C = 8\pi r h + 6\pi r^2 \] --- ## **Step 4: Express Cost in Single Variable** From step 2: \[ h = \frac{28.875}{\pi r^2} \] Substitute \( h \) into cost function: \[ C(r) = 8\pi r \left(\frac{28.875}{\pi r^2}\right) + 6\pi r^2 \] \[ C(r) = 8 \cdot 28.875 \frac{1}{r} + 6\pi r^2 \] \[ C(r) = 231 \frac{1}{r} + 6\pi r^2 \] --- ## **Step 5: Derivative of Cost Function** \[ C(r) = \frac{231}{r} + 6\pi r^2 \] \[ C'(r) = -231r^{-2} + 12\pi r \] Set \( C'(r) = 0 \) to find the minimum: \[ -231r^{-2} + 12\pi r = 0 \] \[ 12\pi r = \frac{231}{r^2} \] \[ 12\pi r^3 = 231 \] \[ r^3 = \frac{231}{12\pi} \] \[ r = \left(\frac{231}{12\pi}\right)^{1/3} \] --- ## **Step 6: Second Derivative Test** \[ C''(r) = 462r^{-3} + 12\pi \] Since both terms are positive for \( r > 0 \), the function is concave up, confirming a minimum. --- ## **Step 7: Find the Radius and Height** \[ r = \left(\frac{231}{12\pi}\right)^{1/3} \] Calculate numerically: - \( 12\pi \approx 37.6991 \) - \( 231 / 37.6991 \approx 6.129 \) - \( r = 6.129^{1/3} \approx 1.838 \) inches Now, find \( h \): \[ h = \frac{28.875}{\pi r^2} \] \[ r^2 \approx (1.838)^2 \approx 3.378 \] \[ h = \frac{28.875}{\pi \cdot 3.378} \] \[ h \approx \frac{28.875}{10.616} \] \[ h \approx 2.72 \text{ inches} \] --- ## **Step 8: Final Answers** **Radius that minimizes cost:** \[ \boxed{1.84 \text{ inches}} \] **Height that minimizes cost:** \[ \boxed{2.72 \text{ inches}} \] --- **Summary Table:** | Quantity | Value (rounded to 2 decimal places) | |----------|-------------------------------------| | Radius | 1.84 inches | | Height | 2.72 inches | Let me know if you need the answer in exact form or further clarification!