# Applied maximum and minimum problems

1. Oct 27, 2010

### donutmax

Q: A box will be built with a square base and an open top. Material for the base costs $8 per square foot, while material for the sides costs$2 per square foot. Find the dimensions of the box of maximum volume that can be built for $2400. A:$8$$=c1/b$$
\$2$$=c2/s$$
$$C=c1+4c2=8b+(4)(2)s=8b+8s=8x^{2}+8xy=8(x^{2}+xy)=2400$$
$$V=x^{2}y$$

$$D_{x}C=?:$$
$$D_{x}C=8x[2x+y+xD_{x}y]=0$$

$$D_{x}y=?:$$
$$2x+y+xD_{x}y=0$$
$$xD_{x}y=-2x-y$$
$$D_{x}y=-2-y/x$$

$$D_{x}V=?:$$
$$D_{x}V=2xy+x^{x}D_{x}y=2xy-(2+y/x)x^{2}$$
$$=2xy-2x^{2}-yx$$
$$=xy-2x^{2}$$

$$D_{x}V=0:$$
$$0=xy-2x^{2}$$
$$2x^{2}=xy$$
$$2x=y$$

$$x^{2}+xy=300$$
$$x(x+y)=300$$
$$2x=y$$
$$x(x+2x)=x(3x)=300$$
$$x^{2}=100$$
$$x=10$$

x | 0 _ 10
y | 300 0 20
V | 0 0 2000

Therefore x=10, y=20

Chk $$D^{2}_{x}V<0:$$
$$D^{2}_{x}V=y+xD_{x}y-4x=y-(2+y/x)x-4x<0$$ when x=10, y=20
=> abs max

Last edited: Oct 27, 2010