Questions: Select the correct answer. Shawn is packing these two boxes with food collected during his school's canned food drive. Which expression represents the total volume of the two boxes? 84x^3-282x^2-138x-336 13x^2+20x-57 21x^3-48x^2-199x 27x^3+48x^2-99x

Select the correct answer.
Shawn is packing these two boxes with food collected during his school's canned food drive.

Which expression represents the total volume of the two boxes?
84x^3-282x^2-138x-336
13x^2+20x-57
21x^3-48x^2-199x
27x^3+48x^2-99x

Transcript text: Select the correct answer. Shawn is packing these two boxes with food collected during his school's canned food drive. Which expression represents the total volume of the two boxes? $84 x^{3}-282 x^{2}-138 x-336$ $13 x^{2}+20 x-57$ $21 x^{3}-48 x^{2}-199 x$ $27 x^{3}+48 x^{2}-99 x$

Solution

Solution Steps

Step 1: Identify the dimensions of the boxes

The dimensions of the two boxes are given as follows:

Box 1: $ (x + 3) \times 3x \times 2 $
Box 2: $ (4x - 8) \times 2x \times (3x + 9) $

Step 2: Calculate the volume of Box 1

The volume $ V_1 $ of Box 1 is calculated by multiplying its dimensions: \[ V_1 = (x + 3) \times 3x \times 2 \] \[ V_1 = 2 \times 3x \times (x + 3) \] \[ V_1 = 6x(x + 3) \] \[ V_1 = 6x^2 + 18x \]

Step 3: Calculate the volume of Box 2

The volume $ V_2 $ of Box 2 is calculated by multiplying its dimensions: \[ V_2 = (4x - 8) \times 2x \times (3x + 9) \] \[ V_2 = 2x \times (4x - 8) \times (3x + 9) \] First, expand $ (4x - 8)(3x + 9) $: \[ (4x - 8)(3x + 9) = 4x \cdot 3x + 4x \cdot 9 - 8 \cdot 3x - 8 \cdot 9 \] \[ = 12x^2 + 36x - 24x - 72 \] \[ = 12x^2 + 12x - 72 \] Now, multiply by $ 2x $: \[ V_2 = 2x(12x^2 + 12x - 72) \] \[ = 24x^3 + 24x^2 - 144x \]

Step 4: Add the volumes of the two boxes

The total volume $ V $ is the sum of $ V_1 $ and $ V_2 $: \[ V = V_1 + V_2 \] \[ V = (6x^2 + 18x) + (24x^3 + 24x^2 - 144x) \] Combine like terms: \[ V = 24x^3 + 30x^2 - 126x \]