In here, why cross multiplication not allowed?
SA = 2(lw+wh+ lh)
V = lwh
We are given that the length is 3, height is h, and width is w.
SA = 2(lw+wh+ lh) = 2(3w+wh+3h) = 6w+2wh+6h
V = lwh = 3wh.
Since these are equal, we get that 6w+2wh+6h = 3wh.
Let’s solve for h in terms of w.
6h - wh = -6w
h(6-w) = -6w
h = 6w/(w-6) (I flipped the terms in the denominator to distribute the negative sign)
You can plug in numbers and see that 6w/(w-6) is going to be greater than 6 for w > 6. If w = 6, then we get that h(6-6) = -6x6 → h(0) = -36 → 0 = -36. This is a contradiction, so w cannot be equal to 6. If w < 6, then the height is negative, which is not allowed.
For an algebraic proof, let’s check when 6w/(w-6) > 6.
6w/(w-6) > 6 → 6w/(w-6) - 6 >0 → 6w/(w-6) - (6w-36)/(w-6) > 0 → (6w - 6w + 36) / (w-6) > 0 → 36/(w-6) > 0. In order for this to be true, the denominator has to be positive, so w - 6 >0 → w > 6. Thus, for w > 6, h = 6w/(w-6) > 6. Thus, we can say that Quantity A is greater than Quantity B.
Cross multiplication would be doing the reverse of solving for h. We want to get h by itself to try different w values.