Ask a New Question
Menu

The ratio of side lengths for two similar cubes is 2/5 . Determine the ratios for each of the following. Show your work or explain how you got your answers. the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube the total surface area of the smaller cube compared to the total surface area of the larger cube the volume of the smaller cube compared to the volume of the larger cube

1 answer

To solve the problem regarding the ratios of various properties of two similar cubes with a side length ratio of \( \frac{2}{5} \), we will first denote the side lengths of the smaller and larger cubes.

Let:

  • \( s_1 \) = side length of the smaller cube
  • \( s_2 \) = side length of the larger cube

Given that: \[ \frac{s_1}{s_2} = \frac{2}{5} \]

Perimeter of One Face

The perimeter of a square face of a cube is calculated using the formula: \[ \text{Perimeter} = 4 \times \text{side length} \]

Thus, for the smaller cube, the perimeter \( P_1 \) is: \[ P_1 = 4 \times s_1 \]

For the larger cube, the perimeter \( P_2 \) is: \[ P_2 = 4 \times s_2 \]

Now, we can find the ratio of the perimeters: \[ \frac{P_1}{P_2} = \frac{4 \times s_1}{4 \times s_2} = \frac{s_1}{s_2} = \frac{2}{5} \]

Total Surface Area

The total surface area \( A \) of a cube is calculated using the formula: \[ \text{Surface Area} = 6 \times (\text{side length})^2 \]

So, the surface area of the smaller cube \( A_1 \) is: \[ A_1 = 6 \times s_1^2 \]

For the larger cube, the surface area \( A_2 \) is: \[ A_2 = 6 \times s_2^2 \]

Now, we find the ratio of the surface areas: \[ \frac{A_1}{A_2} = \frac{6 \times s_1^2}{6 \times s_2^2} = \frac{s_1^2}{s_2^2} \]

Since \( \frac{s_1}{s_2} = \frac{2}{5} \), we can square this ratio: \[ \frac{s_1^2}{s_2^2} = \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]

Volume

The volume \( V \) of a cube is given by: \[ \text{Volume} = (\text{side length})^3 \]

For the smaller cube, the volume \( V_1 \) is: \[ V_1 = s_1^3 \]

For the larger cube, the volume \( V_2 \) is: \[ V_2 = s_2^3 \]

The ratio of the volumes is: \[ \frac{V_1}{V_2} = \frac{s_1^3}{s_2^3} \]

Taking the cube of the ratio of side lengths: \[ \frac{s_1^3}{s_2^3} = \left(\frac{2}{5}\right)^3 = \frac{8}{125} \]

Summary

  1. Ratio of the perimeter of one face of the smaller cube to the larger cube: \[ \frac{P_1}{P_2} = \frac{2}{5} \]

  2. Ratio of the total surface area of the smaller cube to the larger cube: \[ \frac{A_1}{A_2} = \frac{4}{25} \]

  3. Ratio of the volume of the smaller cube to the larger cube: \[ \frac{V_1}{V_2} = \frac{8}{125} \]