Question

9. The polynomial -x^3 -x^2 + 12x represents the volume of a rectangular aquatic tank in cubic feet. The length of the tank is (x + 4).

a. How many linear factors should you look for?

b. What are the dimensions of the tank?

c. Find the value of x that will maximize the volume of the box.

Answer 1

a) because of the x^3 terms, we should expect 3 linear factors, besides, volume is length x width x height

b) V = -x^3 -x^2 + 12x
= x(-x^2 - x +12)
we are told one of the factors is x+4
= x(x+4)(3-x)
so the dimensions of the tank are x by x+4 by 3-x

c) V = -x^3 -x^2 + 12x
dV/dx = -3x^2 - 2x + 12 = 0 for a max/min of V
3x^2 + 2x - 12 = 0
x = appr 1.69 or a negative.

the max volume = -(1.69)^3 - (1.69)^2 + 12(1.69) = 12.60 units^3

looks good, http://www.wolframalpha.com/input/?i=plot+y+%3D++-x%5E3+-x%5E2+%2B+12x