What is the answer, correct to one decimal place, for the following problem?You are given a piece of sheet metal that is twice as long as it is...

The area of the sheet metal is 800 m^2. The length is
twice as long as the the width. Let the length be L. If the width is W, L =
2W.

800 = 2*W*W

=> W =
sqrt(400)

=> W = 20

L =
40

Four squares are cut from the corners of the sheet of 40
x 20 cm and a box is constructed without a lid. Let the squares that are cut have a side
of length s.

The volume of the box is : V = L*W*H, where L,
W and H are the length, width and height of the the
box.

The Length is (40 -
2s)

Width = 20 - 2s

Height  =
s

Volume = (40 - 2s)(20 -
2s)(s)

=> (40s - 2s^2)(20 -
2s)

=> 800s - 40s^2 - 80s^2 +
4s^3

=> 4s^3 - 120s^2 +
800s

To maximize volume find the derivative and solve for
s.

12s^2 - 240s + 800 =
0

=> 3s^2 - 60s + 200 =
0

s1 = 15.77

s2 =
4.22

There are two roots of the equation, one giving a
minimum point and the other the maximum point. The second derivative of the volume which
is 24s - 240 is used to determine which of the values of s give the maximum. 24s - 240
has a negative value for the value of s that gives a maximum
value.

For s = 15.7, 24s - 240 =
138.48

and for s = 4.2 , 24s - 240 =
-138.72

This indicates that volume is minimum if s = 15.77
and maximum if s = 4.2.

For s = 4.2, the dimensions of the
box are: 31.6, 11.6 and 4.2

The box has a
maximum volume when the dimensions are 31.6, 11.6 and 4.2
cm