Question

a clothing company wants to ship more items per box. the length, width, and height of the new boxes are all twice as long as those of the old boxes. how does the volume of a new box compare to the volume of an old box?

a. the volume of the new box is 2 times the volume of the old box.

b. the volume of the new box is 4 times the volume of the old box.

c. the volume of the new box is 6 times the volume of the old box.

d. the volume of the new box is 8 times the volume of the old box.

Explanation:

Step1: Recall the volume formula for a rectangular box

The volume \( V \) of a rectangular box with length \( l \), width \( w \), and height \( h \) is given by \( V = l \times w \times h \).

Step2: Define the dimensions of the old and new boxes

Let the length, width, and height of the old box be \( l \), \( w \), and \( h \) respectively. Then the length, width, and height of the new box will be \( 2l \), \( 2w \), and \( 2h \) (since each dimension is twice as long).

Step3: Calculate the volume of the old box

The volume of the old box, \( V_{\text{old}} = l \times w \times h \).

Step4: Calculate the volume of the new box

The volume of the new box, \( V_{\text{new}} = (2l) \times (2w) \times (2h) \).
Simplify this expression:
\[

$$\begin{align*} V_{\text{new}}&= 2l \times 2w \times 2h\\ &= 2\times2\times2\times l\times w\times h\\ &= 8\times(l\times w\times h) \end{align*}$$

\]
But \( l \times w \times h = V_{\text{old}} \), so \( V_{\text{new}} = 8V_{\text{old}} \).

Answer:

D. The volume of the new box is 8 times the volume of the old box.