Question

How do you solve this?

A warehouse is packing materials to ship out to a customer. A small box has the dimensions of x inches, (x+ 2) inches and (2x) inches. Where x is the length in inches of width. A large box has the dimensions of (x+5) inches, (x + 7) inches and (3x) inches.

*Recall that Volume = (Length)(Width)(Height)

Part 1: Write an expression that represents the dimensions of the small box.

Part 2: Write an expression that represents the dimensions of the large box.

Part 3: What is the difference in the volumes of the two boxes? Show or explain or work.

(I think I understand parts 1 and 2 but I'm definitely struggling with part 3)

Answer 1

The difference in volume is `x(x^2 + 32x + 105)` cubic inches.

Explanation:

I'm going to do all three parts just so that you can compare your work.

`V_"small" = x(2x)(x +2) = x(2x^2 + 4x) = 2x^3 + 4x^2`

`V_"large" = (x + 5)(x + 7)(3x) = (x^2 + 12x + 35)(3x) = 3x^3 + 36x^2 + 105x`

Now, subtract the largest volume from the small volume to get the difference in volume. Call the difference `D`.

`D = V_"large" - V_"small"`

`D = 3x^3 + 36x^2 + 105x - (2x^3 + 4x^2)`

`D = 3x^3 + 36x^2 + 105x - 2x^3 - 4x^2`

`D = x^3 + 32x^2 + 105x`

`D = x(x^2 + 32x + 105)`

This cannot be factored any further.

Hopefully this helps!