To find the amount of icing needed to cover the sides and top of a cylindrical cake, we will calculate the surface area of the cylinder (which consists of the lateral area and the area of the top face).
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Lateral Surface Area (sides): The formula for the lateral surface area \( A_{\text{lateral}} \) of a cylinder is: \[ A_{\text{lateral}} = 2 \pi r h \] where \( r \) is the radius and \( h \) is the height.
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Area of the Top Face: The formula for the area \( A_{\text{top}} \) of the top face of the cylinder is: \[ A_{\text{top}} = \pi r^2 \]
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Total Surface Area: The total surface area \( A_{\text{total}} \) that needs icing will be the sum of the lateral surface area and the area of the top face: \[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{top}} \]
Now let’s calculate each component:
Step 1: Calculate the Lateral Surface Area
Given:
- \( r = 5 \) inches
- \( h = 7 \) inches
\[ A_{\text{lateral}} = 2 \times 3.14 \times 5 \times 7 \] \[ A_{\text{lateral}} = 2 \times 3.14 \times 35 \] \[ A_{\text{lateral}} = 6.28 \times 35 \] \[ A_{\text{lateral}} = 219.8 \text{ square inches} \]
Step 2: Calculate the Area of the Top Face
\[ A_{\text{top}} = 3.14 \times (5^2) \] \[ A_{\text{top}} = 3.14 \times 25 \] \[ A_{\text{top}} = 78.5 \text{ square inches} \]
Step 3: Calculate the Total Surface Area
\[ A_{\text{total}} = 219.8 + 78.5 \] \[ A_{\text{total}} = 298.3 \text{ square inches} \]
Finally, rounding to the nearest tenth, the total amount of icing needed to cover the sides and top of the cake is: \[ \boxed{298.3} \text{ square inches} \]