Question

zoe is packaging a gift to send in the mail. a smaller gift box is placed inside a similar yet larger packaging box. the empty space inside the packaging box is filled with tissue paper to keep the gift box from moving. the dimensions of the larger box are each 2.1 times longer than the corresponding dimensions of the smaller box. the volume of the smaller box is 24 cubic centimeters. what is the volume of the space filled with tissue paper in the packaging box? 81.84 cubic centimeters 105.84 cubic centimeters 198.264 cubic centimeters 222.264 cubic centimeters

Explanation:

Step1: Recall volume - ratio for similar solids

If the ratio of corresponding side - lengths of two similar solids is \(k\), the ratio of their volumes is \(k^{3}\). Here, \(k = 2.1\).

Step2: Calculate the volume of the larger box

Let \(V_1\) be the volume of the smaller box and \(V_2\) be the volume of the larger box. We know that \(\frac{V_2}{V_1}=k^{3}\), and \(V_1 = 24\) cubic centimeters. So \(V_2=V_1\times k^{3}=24\times(2.1)^{3}\).
\[

$$\begin{align*} (2.1)^{3}&=2.1\times2.1\times2.1\\ & = 9.261 \end{align*}$$

\]
Then \(V_2=24\times9.261 = 222.264\) cubic centimeters.

Step3: Calculate the volume of the tissue - paper

The volume of the tissue - paper \(V = V_2 - V_1\). Since \(V_2 = 222.264\) cubic centimeters and \(V_1 = 24\) cubic centimeters, \(V=222.264 - 24=198.264\) cubic centimeters.

Answer:

198.264 cubic centimeters