Problema Solution
A model of carry-on luggage has a length that is 6 inches greater than depth. Airline regulations require that the sum of the length, width, and depth cannot exceed 24 inches. These conditions, with the assumption that this sum is 24 inches, can be modeled by the function V(x) that gives the luggage's volume, in cubic inches, in terms of its depth, x, in inches. If its volume is 432 cubic inches, determine two possibilities for its depth.
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debth = x
length = x + 6
width = 24 - (x + x + 6)
V(x) = debth * length * width
V(x) = x * (x+6) * [24-(x+x+6)]
V(x) = x(x+6)(18-2x)
the volume is 432 cubic inches meaning
x(x+6)(18-2x) = 432 divide both sides by 2
x(x+6)(9-x) = 216
-x^3 + 9x^3 - 6x^2 + 54x - 216 = 0
one possibility for debth is x = 6 indeed 6*12*3 = 216
divide -x^3 + 9x^3 - 6x^2 + 54x - 216 = (x - 6)(-x^2 - 3x + 36) we have
(x - 6)(-x^2 - 3x + 36) = 0
by solving
-x^2 - 3x + 36 = 0
we find another possibility for debth
x = 4.68 in
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two possibilities for its depth are 6 in and 4.68 in.