Question

one cube has edges n meters long. another has edges 3n meters long. what is the ratio of the volume of the first cube to the volume of the second cube? a. 1:3 b. 1:9 c. 1:27 d. 1:6 e. 1:81

Explanation:

Step1: Recall volume formula for cube

The volume formula of a cube is $V = s^3$, where $s$ is the edge - length. For the first cube with edge - length $n$, its volume $V_1=n^3$. For the second cube with edge - length $3n$, its volume $V_2=(3n)^3$.

Step2: Expand the volume of the second cube

Using the power - of - a - product rule $(ab)^n=a^n b^n$, we have $(3n)^3 = 3^3\times n^3=27n^3$.

Step3: Find the ratio of the volumes

The ratio of the volume of the first cube to the volume of the second cube is $\frac{V_1}{V_2}=\frac{n^3}{27n^3}$. Canceling out the $n^3$ terms, we get $\frac{1}{27}$, which is equivalent to $1:27$.

Answer:

C. $1:27$