To calculate the surface area of the cylindrical cake, we need to find the surface area of the cylinder, which includes two parts: the lateral surface area and the area of the top and bottom.
The formulas for surface area are as follows:
- Lateral Surface Area (A_lateral) = 2πrh
- Area of the top and bottom (A_top_bottom) = 2πr²
Where:
- r = radius of the cylinder
- h = height of the cylinder
- π (pi) is approximately 3.14
Given:
- The diameter of the cake pan is 13 inches, so the radius (r) is half of the diameter:
r = 13 / 2 = 6.5 inches - The height (h) of the cake is 4 inches.
Now we can calculate each part:
-
Lateral Surface Area: \[ A_{lateral} = 2 \cdot \pi \cdot r \cdot h = 2 \cdot 3.14 \cdot 6.5 \cdot 4 \] \[ A_{lateral} = 2 \cdot 3.14 \cdot 26 = 2 \cdot 81.64 = 163.28 \text{ inches}^2 \]
-
Area of the top and bottom: \[ A_{top_bottom} = 2 \cdot \pi \cdot r^2 = 2 \cdot 3.14 \cdot (6.5)^2 \] \[ (6.5)^2 = 42.25 \] \[ A_{top_bottom} = 2 \cdot 3.14 \cdot 42.25 = 6.28 \cdot 42.25 = 265.33 \text{ inches}^2 \]
-
Total Surface Area (A_total): \[ A_{total} = A_{lateral} + A_{top_bottom} = 163.28 + 265.33 = 428.61 \text{ inches}^2 \]
Thus, the total surface area Marco needs to frost is 428.61 inches².
The correct response is 428.61 inches squared.