To find the amount of icing needed to cover the sides and the top of a cylindrical cake, we need to calculate the surface area of the sides and the area of the top of the cylinder.
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Calculate the area of the top: The area \( A \) of a circle (the top of the cake) is given by the formula: \[ A = \pi r^2 \]
Here, the radius \( r \) is 5 inches. Using \( \pi \approx 3.14 \): \[ A = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \text{ square inches} \]
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Calculate the lateral (side) surface area: The lateral surface area \( A_L \) of a cylinder is given by the formula: \[ A_L = 2\pi rh \]
Here, \( h \) is the height of the cake, which is 7 inches: \[ A_L = 2 \times 3.14 \times 5 \times 7 \] Calculating it step by step: \[ A_L = 2 \times 3.14 = 6.28 \] \[ A_L = 6.28 \times 5 = 31.4 \] \[ A_L = 31.4 \times 7 = 219.8 \text{ square inches} \]
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Total icing area needed: Now, we add the area of the top and the lateral surface area: \[ \text{Total area} = A + A_L = 78.5 + 219.8 = 298.3 \text{ square inches} \]
Thus, the total amount of icing needed to cover the sides and the top of the cake is: \[ \boxed{298.3} \text{ square inches} \]
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