To determine the total number of pieces of cake and the side length of each square piece, we first need to find the area of the cake and then calculate the area of each piece based on the percentage given.
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Calculate the area of the cake:
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The dimensions of the cake are \( 13 \frac{1}{2} \) inches and \( 6 \frac{3}{4} \) inches.
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Convert these mixed numbers to improper fractions:
- \( 13 \frac{1}{2} = \frac{27}{2} \) inches
- \( 6 \frac{3}{4} = \frac{27}{4} \) inches
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Calculate the area: \[ \text{Area of the cake} = \left(\frac{27}{2}\right) \times \left(\frac{27}{4}\right) = \frac{27 \times 27}{2 \times 4} = \frac{729}{8} \text{ square inches} \]
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Calculate the area of each piece based on the percentage:
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Each piece is to be \( 5.55% \) of the total cake area.
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Convert the percentage to a decimal: \[ 5.55% = 0.0555 \]
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Calculate the area of each piece: \[ \text{Area of each piece} = 0.0555 \times \frac{729}{8} = 0.0555 \times 91.125 = 5.06 \text{ square inches} \]
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Calculate the side length of each square piece:
- For a square, the area is given by \( s^2 \) where \( s \) is the side length: \[ s^2 = 5.06 \implies s = \sqrt{5.06} \approx 2.25 \text{ inches} \]
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Calculate the total number of pieces:
- The total number of pieces is the total area of the cake divided by the area of each piece: \[ n = \frac{\text{Total Area}}{\text{Area of each piece}} = \frac{\frac{729}{8}}{5.06} \approx \frac{91.125}{5.06} \approx 18 \]
In summary:
- Number of pieces of cake: approximately 18 pieces
- Side length of each piece: approximately 2.25 inches.
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