To find the surface area of a similar shoe box with a volume of 27 cubic inches, we start by establishing the relationship between the volume and surface area of similar three-dimensional objects.
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The volume ratio of similar objects is equal to the cube of the ratio of their corresponding linear dimensions. Let \( k \) be the ratio of the linear dimensions (length, width, height) of the new shoe box to the original shoe box.
\[ \frac{V_2}{V_1} = k^3 \]
Given:
- \( V_1 = 64 \) cubic inches (original shoe box volume)
- \( V_2 = 27 \) cubic inches (new shoe box volume)
\[ \frac{27}{64} = k^3 \]
To find \( k \), we take the cube root of both sides:
\[ k = \sqrt[3]{\frac{27}{64}} = \frac{3}{4} \]
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The surface area ratio of similar objects is equal to the square of the ratio of their corresponding linear dimensions.
\[ \frac{SA_2}{SA_1} = k^2 \]
Given:
- \( SA_1 = 160 \) square inches (original shoe box surface area)
Calculate \( k^2 \):
\[ k^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]
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Now we can find the new surface area \( SA_2 \):
\[ SA_2 = SA_1 \times k^2 = 160 \times \frac{9}{16} \]
Simplifying this gives:
\[ SA_2 = 160 \times \frac{9}{16} = 10 \times 9 = 90 \text{ square inches} \]
Thus, the surface area of the similar shoe box with a volume of 27 cubic inches is 90 square inches.