Question

the ratio of the edge lengths of two cubes is 5:7. (a) find the ratio of their volumes. (b) find the ratio of their surface areas. write these ratios in the format m:n. volume: surface area:

Explanation:

Step1: Recall volume formula for cube

The volume formula of a cube is $V = s^{3}$, where $s$ is the edge - length. Let the edge - lengths of the two cubes be $s_1$ and $s_2$ such that $\frac{s_1}{s_2}=\frac{5}{7}$. The volumes $V_1=s_1^{3}$ and $V_2 = s_2^{3}$. Then $\frac{V_1}{V_2}=\frac{s_1^{3}}{s_2^{3}}=(\frac{s_1}{s_2})^{3}$.

Step2: Calculate volume ratio

Substitute $\frac{s_1}{s_2}=\frac{5}{7}$ into the volume - ratio formula. $\frac{V_1}{V_2}=(\frac{5}{7})^{3}=\frac{125}{343}$.

Step3: Recall surface - area formula for cube

The surface - area formula of a cube is $A = 6s^{2}$. Let the surface - areas of the two cubes be $A_1 = 6s_1^{2}$ and $A_2=6s_2^{2}$. Then $\frac{A_1}{A_2}=\frac{6s_1^{2}}{6s_2^{2}}=\frac{s_1^{2}}{s_2^{2}}$.

Step4: Calculate surface - area ratio

Substitute $\frac{s_1}{s_2}=\frac{5}{7}$ into the surface - area ratio formula. $\frac{A_1}{A_2}=(\frac{5}{7})^{2}=\frac{25}{49}$.

Answer:

(a) $125:343$
(b) $25:49$