Bot GPT 4-o mini To find the total area of icing needed to cover the sides and the top of the cylindrical cake, we need to calculate the lateral surface area and the area of the top surface of the cylinder.
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Lateral Surface Area: The formula for the lateral surface area of a cylinder is: \[ A_{lateral} = 2\pi rh \] where \( r \) is the radius and \( h \) is the height.
Substituting the given values:
- Radius \( r = 5 \) inches
- Height \( h = 7 \) inches
\[ A_{lateral} = 2 \times 3.14 \times 5 \times 7 \]
Calculating it step by step: \[ A_{lateral} = 2 \times 3.14 \times 5 = 31.4 \] \[ A_{lateral} = 31.4 \times 7 = 219.8 \text{ square inches} \]
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Area of the Top Surface: The area of the top surface of the cylinder (which is a circle) is given by the formula: \[ A_{top} = \pi r^2 \]
Substituting for the radius: \[ A_{top} = 3.14 \times (5^2) = 3.14 \times 25 = 78.5 \text{ square inches} \]
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Total Area of Icing Needed: The total area that needs icing is the sum of the lateral surface area and the top surface area: \[ A_{total} = A_{lateral} + A_{top} \] \[ A_{total} = 219.8 + 78.5 = 298.3 \text{ square inches} \]
Rounding to the nearest tenth, the total icing needed is: \[ \boxed{298.3} \text{ square inches} \]
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