An open box is constructed by cutting congruent squares from the corners of a 30 inch by 20 inch piece of aluminum.

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(20-2x)(30-2x)= 600x -100x^2 +4x^3

v'= 600 -200x +12x^2 = 0

3x^2 -50x +150 = 0

x= 50/6 +/- (1/6)sqr(2500 -1800)

x = (50 - 10sqr7)/6 = about 3.924 inches = height of the box

or (50+10sqr7)/6 = about 12.743 inches which is not possible

x= about 3.924 inches by 3.924 inches cut from each of the 4 corners

sides of the box are 20-2(3.924) = 12.152 inches

and 30-2(3.924) = 22.152 inches

Maximum Volume = 3.924(12.152)(22.152)

= about 1,056.306 cubic inches