Question

a shipping company sells large boxes and small boxes. the lengths of the sides of the small box are $\frac{1}{3}$ of the lengths of the large box. which statement is true? a. the volume of the small box is $\frac{1}{3}$ of the volume of the large box. b. the volume of the small box is $\frac{1}{9}$ of the volume of the large box. c. the volume of the small box is $\frac{1}{27}$ of the volume of the large box. d. the volume of the small box is $\frac{1}{81}$ of the volume of the large box.

Explanation:

Step1: Recall volume formula for box

The volume formula for a box (rectangular - prism) is $V = l\times w\times h$. Let the side - lengths of the large box be $l_1$, $w_1$, and $h_1$, so its volume $V_1=l_1\times w_1\times h_1$. The side - lengths of the small box are $\frac{1}{3}l_1$, $\frac{1}{3}w_1$, and $\frac{1}{3}h_1$.

Step2: Calculate volume of small box

The volume of the small box $V_2=\frac{1}{3}l_1\times\frac{1}{3}w_1\times\frac{1}{3}h_1=\frac{1}{27}(l_1\times w_1\times h_1)$. Since $V_1 = l_1\times w_1\times h_1$, then $V_2=\frac{1}{27}V_1$.

Answer:

C. The volume of the small box is $\frac{1}{27}$ of the volume of the large box.