Let's break down the problem from the image: ### **Given:** - Efficiency of fin, \( \eta = 56\% = .56 \) - Diameter of the fin, \( d = 25 \) cm - Effectiveness of fin, \( \epsilon = 3 \) - Required: Total surface area of the fin (assuming insulated tip). --- ## **Step 1: Formula for Effectiveness** Effectiveness of a fin is given by: \[ \epsilon = \frac{\text{Heat transfer by the fin}}{\text{Heat transfer by the same area without the fin}} \] For a straight fin: \[ \epsilon = \frac{\eta \cdot A_{\text{fin}}}{A_{\text{base}}} \] Where: - \( A_{\text{fin}} \) = surface area of the fin - \( A_{\text{base}} \) = base area = area covered by the base of the fin (here, a circle) --- ## **Step 2: Rearranging for Surface Area** \[ \epsilon = \frac{\eta \cdot A_{\text{fin}}}{A_{\text{base}}} \implies A_{\text{fin}} = \frac{\epsilon \cdot A_{\text{base}}}{\eta} \] --- ## **Step 3: Base Area Calculation** \[ A_{\text{base}} = \text{Area of a circle} = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 \] \[ r = \frac{25}{2} = 12.5 \text{ cm} \] \[ A_{\text{base}} = \pi \times (12.5)^2 \approx 3.1416 \times 156.25 \approx 490.87 \text{ cm}^2 \] --- ## **Step 4: Surface Area Calculation** \[ A_{\text{fin}} = \frac{3 \times 490.87}{.56} \approx \frac{1472.61}{.56} \approx 2639.48 \text{ cm}^2 \] --- ## **Step 5: Matching with Options** From the options in the image: - **2629.98 cm²** - **2941.13 cm²** - **3495.06 cm²** - **1508.09 cm²** **Closest and correct option:** > **2629.98 cm²** --- ## **Final Answer** ### **The correct option is:** **2629.98 cm²**