A metalworker wants to make an open box from a 30 in. x 24x sheet of metal, by cutting equal squares from each corner.

Question

Write a function that models the volume of the box.

What is the maximum volume that the box can hold?

What is the height of the box for the maximum volume?

Answer 1

V(x) = x(30-2x)(24-2x)

= x(720 - 108x + 4x^2)

= 720x -108x^2 +4x^3

V'(x) = 720 -216x +12x^2 = 0

3x^2 -54x + 180 = 0

x^2 -18x +60 = 0

x^2 -18x + (9)^2 = -60 + (9)^2

(x -9)^2 = -60 + 81

x-9= +/-sqr21

x = 9 +sqr21 or 9-sqr21

x = 9-sqr21 (ignore the positive square root, as it gives an impossible solution)

x = about 4.17 inches = height of the box with maximum volume

other box sides are 30-2(4.17) = 21.165

and 24-2(4.17) = 15.165

max volume = 4.17(21.165)(15.165)

= about 1338.45

= about 1,338 1/2 cubic inches