Question
The ratio of side lengths for two similar cubes is 2/5 . Determine the ratios for each of the following. Show your work
a. the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube
b. the total surface area of the smaller cube compared to the total surface area of the larger cube
c. the volume of the smaller cube compared to the volume of the larger cube
a. the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube
b. the total surface area of the smaller cube compared to the total surface area of the larger cube
c. the volume of the smaller cube compared to the volume of the larger cube
Answers
GPT-4o mini
To solve these problems, we start by understanding the properties of similar geometric figures, particularly cubes in this case.
Given that the ratio of the side lengths for two similar cubes is \( \frac{2}{5} \), we can denote the side length of the smaller cube as \( s_1 = 2x \) and the side length of the larger cube as \( s_2 = 5x \) for some variable \( x \).
### a. Ratio of the perimeter of one face
The perimeter \( P \) of a face of a cube is given by the formula:
\[
P = 4 \times \text{side length}
\]
For the smaller cube:
\[
P_1 = 4s_1 = 4(2x) = 8x
\]
For the larger cube:
\[
P_2 = 4s_2 = 4(5x) = 20x
\]
Thus, the ratio of the perimeter of the smaller cube to the perimeter of the larger cube is:
\[
\text{Ratio} = \frac{P_1}{P_2} = \frac{8x}{20x} = \frac{8}{20} = \frac{2}{5}
\]
### b. Ratio of the total surface area
The total surface area \( A \) of a cube is given by the formula:
\[
A = 6 \times (\text{side length})^2
\]
For the smaller cube:
\[
A_1 = 6s_1^2 = 6(2x)^2 = 6 \times 4x^2 = 24x^2
\]
For the larger cube:
\[
A_2 = 6s_2^2 = 6(5x)^2 = 6 \times 25x^2 = 150x^2
\]
Thus, the ratio of the total surface area of the smaller cube to the total surface area of the larger cube is:
\[
\text{Ratio} = \frac{A_1}{A_2} = \frac{24x^2}{150x^2} = \frac{24}{150} = \frac{4}{25}
\]
### c. Ratio of the volume
The volume \( V \) of a cube is given by the formula:
\[
V = (\text{side length})^3
\]
For the smaller cube:
\[
V_1 = s_1^3 = (2x)^3 = 8x^3
\]
For the larger cube:
\[
V_2 = s_2^3 = (5x)^3 = 125x^3
\]
Thus, the ratio of the volume of the smaller cube to the volume of the larger cube is:
\[
\text{Ratio} = \frac{V_1}{V_2} = \frac{8x^3}{125x^3} = \frac{8}{125}
\]
### Summary of the Ratios
a. The perimeter of the smaller cube to the larger cube is \( \frac{2}{5} \).
b. The total surface area of the smaller cube to the larger cube is \( \frac{4}{25} \).
c. The volume of the smaller cube to the larger cube is \( \frac{8}{125} \).
Given that the ratio of the side lengths for two similar cubes is \( \frac{2}{5} \), we can denote the side length of the smaller cube as \( s_1 = 2x \) and the side length of the larger cube as \( s_2 = 5x \) for some variable \( x \).
### a. Ratio of the perimeter of one face
The perimeter \( P \) of a face of a cube is given by the formula:
\[
P = 4 \times \text{side length}
\]
For the smaller cube:
\[
P_1 = 4s_1 = 4(2x) = 8x
\]
For the larger cube:
\[
P_2 = 4s_2 = 4(5x) = 20x
\]
Thus, the ratio of the perimeter of the smaller cube to the perimeter of the larger cube is:
\[
\text{Ratio} = \frac{P_1}{P_2} = \frac{8x}{20x} = \frac{8}{20} = \frac{2}{5}
\]
### b. Ratio of the total surface area
The total surface area \( A \) of a cube is given by the formula:
\[
A = 6 \times (\text{side length})^2
\]
For the smaller cube:
\[
A_1 = 6s_1^2 = 6(2x)^2 = 6 \times 4x^2 = 24x^2
\]
For the larger cube:
\[
A_2 = 6s_2^2 = 6(5x)^2 = 6 \times 25x^2 = 150x^2
\]
Thus, the ratio of the total surface area of the smaller cube to the total surface area of the larger cube is:
\[
\text{Ratio} = \frac{A_1}{A_2} = \frac{24x^2}{150x^2} = \frac{24}{150} = \frac{4}{25}
\]
### c. Ratio of the volume
The volume \( V \) of a cube is given by the formula:
\[
V = (\text{side length})^3
\]
For the smaller cube:
\[
V_1 = s_1^3 = (2x)^3 = 8x^3
\]
For the larger cube:
\[
V_2 = s_2^3 = (5x)^3 = 125x^3
\]
Thus, the ratio of the volume of the smaller cube to the volume of the larger cube is:
\[
\text{Ratio} = \frac{V_1}{V_2} = \frac{8x^3}{125x^3} = \frac{8}{125}
\]
### Summary of the Ratios
a. The perimeter of the smaller cube to the larger cube is \( \frac{2}{5} \).
b. The total surface area of the smaller cube to the larger cube is \( \frac{4}{25} \).
c. The volume of the smaller cube to the larger cube is \( \frac{8}{125} \).