Mth-143 Reducing a cereal box’s volume by 15%

Question

Consider reducing two, but not three of the dimensions by the same number of cm. What would the new dimensions be in each case?

The original dimensions are 30x24x6

The original volume is 4320

The new dimensions must equal 3672

Answer 1

We have a 30x24x6 box, whose volume is 4320. We want to reduce this volume by 15% such that the new volume is 3672. The problem states that in order to achieve this volume reduction, only two out of the three sides may be reduced, and it also states that they both must be reduced by the same amount.

To calculate the volume of a box, we use the formula Length x Width x Height (LxWxH). If we are reducing two sides only, there are three possible cases:

Case 1: Reduce L and W, keep H the same

Case 2: Reduce L and H, keep W the same

Case 3: Reduce W and H, keep L the same

Case 1:

Let's define "x" as the amount we want to reduce each side of our box. Remember that since the problem states both sides are to be reduced by the same amount, we can use a single variable.

V_new = (L-x)*(W-x)*(H)

3672 = (30-x)*(24-x)*(6), (divide both sides by 6)

612 = (30-x)*(24-x), (multiply/expand the terms)

612 = 720 - 54x + x², (subtract 612 from both sides and rearrange terms)

0 = x² - 54x +108 (solve for x using quadratic formula or calculator, if permitted)

x = 2.08, 51.92 (the second answer does not make sense logically, as we cannot have a negative side length)

Therefore, L = (30 - 2.08) = 27.92, W = (24-2.08) = 21.92, H = 6. LxWxH = 27.92 x 21.92 x 6. We can check our answer by calculating LxWxH.

(27.92)*(21.92)*(6) ≈ 3672, OK

Case 2:

We repeat the above steps, this time keeping W the same and reducing L and H.

V_new = (L-x)*(W)*(H-x)

3672 = (30-x)*(24)*(6-x), (divide both sides by 24)

153 = (30-x)*(6-x), (multiply/expand the terms)

153 = 180 - 36x + x², (subtract 153 from both sides and rearrange terms)

0 = x² - 36x +27 (solve for x using quadratic formula or calculator, if permitted)

x = 0.77, 35.23 (the second answer does not make sense logically, as we cannot have a negative side length)

Therefore, L = (30 - 0.77) = 29.23, W = 24, H = (6 - 0.77) = 5.23. LxWxH = 29.23 x 24 x 5.23.

Case 3:

We repeat the above steps, this time keeping L the same and reducing W and H. I will leave this one as an exercise after performing the first two. Remember you can always check your final answer by calculating LxWxH, such as we did in Case 1.