An open box is to be made out of a 10-inch by 16-inch piece of cardboard

Question

An open box is to be made out of a 10-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Dimentions of the bottom of the box: _____ by _____

Height of the box: _____

Answer 1

v(x)=x(10-2x)(16-2x)

=160x -52x^2 +4x^3

v'(x) = 160 -104x +12x^2 =0

3x^2 -26x +40 - 0

x = 26/6 +/- (1/6)sqr(26^2-480)

x=13/3 +/- (sqr196)/6

x= (26+/-14)/6

x= 12/6 or 40/6

x= 2 or 6 2/3

x = 2 (as 6 2/3 is too large for 10-2x=10-13 1/3 =-3 1/3 inches wide, which gives a negative volume)

x = 2 inches

maximum volume = v(2) = 2(10-4)(16-4) = 2(6)(12)

v(2) = 12(12)

v(2) = 144 cubic inches

bottom of the box dimensions are 12 inches by 6 inches

height = 2 inches

as a rough check, try x=1 and x=3,

two simple integers </> x=2

see if they have less volume

v(1) =1(10-2)(16-2) = 1(8)(14) = 112 in^3 < 144 in^3

v(3) =3(10-6)(16-6) = 3(4)(10) = 120 in^3 < 144 in^3