Question

a rectangular box has a volume of 15 cubic feet. the length, width, and height of the box are each doubled. what is the new volume of the rectangular box?
a. 30 cubic feet
b. 60 cubic feet
c. 90 cubic feet
d. 120 cubic feet

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 \( V = l \times w \times h \). Initially, the volume \( V_1 = l_1 \times w_1 \times h_1 = 15 \) cubic feet.

Step2: Determine the new dimensions

The new length \( l_2 = 2l_1 \), new width \( w_2 = 2w_1 \), and new height \( h_2 = 2h_1 \).

Step3: Calculate the new volume

The new volume \( V_2 = l_2 \times w_2 \times h_2 \). Substituting the new dimensions, we get \( V_2=(2l_1)\times(2w_1)\times(2h_1) \). Using the associative property of multiplication, this can be rewritten as \( V_2 = 2\times2\times2\times l_1\times w_1\times h_1 \). Since \( l_1\times w_1\times h_1 = V_1 = 15 \), we have \( V_2 = 8\times V_1 \)? Wait, no, wait. Wait, \( 2\times2\times2 = 8 \)? Wait, no, wait, no, hold on. Wait, no, the initial volume is \( lwh = 15 \). If we double each dimension, the new volume is \( (2l)(2w)(2h)=8lwh \)? Wait, no, that can't be. Wait, no, wait, no, I made a mistake. Wait, no, the problem says "each doubled", so length becomes \( 2l \), width \( 2w \), height \( 2h \). So the new volume is \( (2l)(2w)(2h)=8lwh \)? But wait, the initial volume is \( lwh = 15 \), so new volume is \( 8\times15 = 120 \)? Wait, but let's check again. Wait, no, wait, no, maybe I messed up. Wait, no, the formula for the volume of a rectangular prism (box) is \( V = l \times w \times h \). If we scale each dimension by a factor of \( k \), the new volume is \( k^3 \) times the original volume. Here, \( k = 2 \), so the new volume is \( 2^3 \times V_1 = 8\times15 = 120 \)? Wait, but let's check the answer options. Option D is 120. Wait, but let's do it step by step again.

Wait, initial volume: \( V_1 = l \times w \times h = 15 \).

New length: \( l' = 2l \)

New width: \( w' = 2w \)

New height: \( h' = 2h \)

New volume: \( V_2 = l' \times w' \times h' = (2l) \times (2w) \times (2h) = 2 \times 2 \times 2 \times l \times w \times h = 8 \times (l \times w \times h) \). Since \( l \times w \times h = 15 \), then \( V_2 = 8 \times 15 = 120 \) cubic feet.

Answer:

D. 120 cubic feet