To calculate the total area of icing needed to cover the sides and top of a cylinder cake, we need to find the lateral surface area and the area of the top of the cylinder.
- Lateral Surface Area (A_lateral): The formula for the lateral surface area of a cylinder is given by:
\[ A_{\text{lateral}} = 2 \pi r h \]
Where:
- \( r \) is the radius (5 inches)
- \( h \) is the height (7 inches)
Using \( \pi \approx 3.14\):
\[ A_{\text{lateral}} = 2 \times 3.14 \times 5 \times 7 \] \[ A_{\text{lateral}} = 2 \times 3.14 \times 35 \] \[ A_{\text{lateral}} = 6.28 \times 35 \] \[ A_{\text{lateral}} = 219.8 \text{ square inches} \]
- Area of the Top (A_top): The formula for the area of the top of the cylinder is given by:
\[ A_{\text{top}} = \pi r^2 \]
Plugging in the radius:
\[ A_{\text{top}} = 3.14 \times (5^2) = 3.14 \times 25 \] \[ A_{\text{top}} = 78.5 \text{ square inches} \]
- Total Area (A_total): Now, we add the lateral area and the top area together:
\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{top}} \] \[ A_{\text{total}} = 219.8 + 78.5 \] \[ A_{\text{total}} = 298.3 \text{ square inches} \]
Thus, the total area of icing needed to cover the sides and top of the cake is approximately 298.3 square inches.