Length of the box = 25 - 2x Width of the box = 14 - 2x Height of the box = x
a.) Volume of the box = x(25 - 2x)(14 - 2x) = x(350 - 50x - 28x + 4x^2) = 4x^3 - 78x^2 + 350x Therefore, the function that models the volume of the box is V(x) = 4x^3 - 78x^2 + 350x
b.) 4x^3 - 78x^2 + 350x ≥ 240 4x^3 - 78x + 350x - 240 ≥ 0 x = 0.834, 5.438 The values of x for which the volume is greater than 240 in^3 is 0.834 ≤ x ≤ 5.438
c.) For maximum volume, dV/dx = 0 dV/dx = 12x^2 - 156x + 350 = 0 x = 2.882910931
