Question

Help please

A graphing calculator is recommended.

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions W = 14 in. by L = 25 in. by cutting out equal squares of side x at each corner and then folding up the sides (see the figure).

(a) Find a function that models the volume V of the box.
V(x) =
4x3−78x2+350


(b) Find the values of x for which the volume is greater than 240 in3.

0.834 ≤ x ≤ 5.438

(c) Find the largest volume that such a box can have. (Round your answer to three decimal places.)

Answer 1

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

Therefore, maximum volume = 4(2.882910931)^3 - 78(2.882910931)^2 +350(2.882910931) = 453.798 in^3