To determine the ratios for perimeter, surface area, and volume of the smaller cube compared to the larger cube using the given ratio of their side lengths (2/5), we can utilize the properties of similar shapes:
Given that the side lengths of the smaller cube (let's call it \( s_1 \)) and the larger cube (let's call it \( s_2 \)) are in the ratio of \( \frac{s_1}{s_2} = \frac{2}{5} \).
a. The perimeter of one face
The perimeter \( P \) of one face of a cube is given by \( P = 4s \), where \( s \) is the length of a side of the cube.
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For the smaller cube: \[ P_{\text{small}} = 4s_1 = 4 \left( \frac{2}{5}s_2 \right) = \frac{8}{5}s_2 \]
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For the larger cube: \[ P_{\text{large}} = 4s_2 \]
Now, we find the ratio of the perimeters: \[ \text{Ratio}{\text{perimeter}} = \frac{P{\text{small}}}{P_{\text{large}}} = \frac{\frac{8}{5}s_2}{4s_2} = \frac{8/5}{4} = \frac{8}{20} = \frac{2}{5} \]
b. The total surface area
The surface area \( A \) of a cube is given by \( A = 6s^2 \).
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For the smaller cube: \[ A_{\text{small}} = 6s_1^2 = 6 \left( \frac{2}{5}s_2 \right)^2 = 6 \left( \frac{4}{25}s_2^2 \right) = \frac{24}{25}s_2^2 \]
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For the larger cube: \[ A_{\text{large}} = 6s_2^2 \]
Now, we find the ratio of the surface areas: \[ \text{Ratio}{\text{surface area}} = \frac{A{\text{small}}}{A_{\text{large}}} = \frac{\frac{24}{25}s_2^2}{6s_2^2} = \frac{24/25}{6} = \frac{24}{150} = \frac{4}{25} \]
c. The volume
The volume \( V \) of a cube is given by \( V = s^3 \).
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For the smaller cube: \[ V_{\text{small}} = s_1^3 = \left( \frac{2}{5}s_2 \right)^3 = \frac{8}{125}s_2^3 \]
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For the larger cube: \[ V_{\text{large}} = s_2^3 \]
Now, we find the ratio of the volumes: \[ \text{Ratio}{\text{volume}} = \frac{V{\text{small}}}{V_{\text{large}}} = \frac{\frac{8}{125}s_2^3}{s_2^3} = \frac{8}{125} \]
Final Ratios
- a. The ratio of the perimeter of one face of the smaller cube to the larger cube is \( \frac{2}{5} \).
- b. The ratio of the total surface area of the smaller cube to the larger cube is \( \frac{4}{25} \).
- c. The ratio of the volume of the smaller cube to the larger cube is \( \frac{8}{125} \).