I am stuck on this question can I get some help

Question

Compare the area of the bases. If a base has a greater area, it will take up more space on the shelf.

Base area of Box 1: 3x²

Base area of Box 2: 4x² – x

Which box is likely to occupy more space on the shelf? Explain. Try substituting different values for x.

Answer 1

Yes - try substituting values for x!

x = 1 3x² = 3(1) = 3 4x² - x = 4(1) - 1 = 3 (no difference)

x = 2 = 3(4) = 12 = 4(4) - 2 = 14

x = 5 = 3(25) = 75 = 4(25) - 5 = 95

x = 10 = 3(100) = 300 = 4(100) - 10 = 390

Looks like the 3x² area will always be smaller except when x = 1. Note: We're dealing with a dimension here, so x can't be 0 or negative.

Answer 2

One way to approach this is to find the zeroes of B₂ - B₁ = 4x² - x - (3x²) = x² - x

This happens when x² = x or x = 0 or 1 (It does not work for -1)

In the interval 0 to 1 the difference in base area is < 0, which means B₁ is greater. For x > 1, B₂ is greater.

We also cannot have a negative area so we set both of the areas to 0 in order to see the limits in the domain:

3x² ≥ 0 means x > 0 and 4x² - x ≥ 0 means x ≥ 1/4 (I've assumed x ≥ 0

From x = 1/4 to 1 B₁ is greater

From x = 1 to infinity B₂ is greater. (B₂ is bigger for more x values) (I think that would be a better problem if the shelf space were given. This would put an upper bound on x and the question would make more sense. (note that if we allow x < 0, there is no limit on either base and B₂ is always larger)

If you do B₂/B₁, that appears clearer: 4/3 - 1/3x This value is always >1 except in the interval 1/4 < x < 1. (disallowed 0 <x < 1/4)