a piece of wire 60 inches long is cut into six sections, two of one length and four of another length. Each of the two sections having the same length is bent into the form of a circle and the two circles are then joined by the four remaining sections to make a frame for a model of a right circular cylinder of, as shown in the accompanying figure.
-find the lengths of the sections that will make the cylinder of maximum volume
- justify your answer
2007-01-04 14:30:37 · 2 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
A very interesting problem.
Let x (in) be the length of the two equal sections. Then the length of the four equal sections would be (60-2x)/4 = 15 - 0.5x in.
Also, we have
2pi r = x, h = 15-0.5x, where r is the radius and h is the height of the cylinder.
Solve for r, r = x/(2 pi)
v = pi r^2 h = pi(x/(2pi))^2(15-0.5x)......(1)
Differentiate (1) with respect to x,
v' = [2x(15-0.5x) - 0.5x^2 ]/(4pi).......(2)
Solve v' = 0 for x,
1.5x^2 - 30x = 0
1.5x(x-20) = 0
Therefore, we have
x = 20 in,
h = 15-0.5x = 5 in
2007-01-04 14:53:32 · answer #1 · answered by sahsjing 7 · 0⤊ 0⤋
volume = pi*r^2*h for a right cylinder
2*(2pi*r)+4h = 60in or h=15-pi*r
Substiute for h in the first equation
2007-01-04 14:35:24 · answer #2 · answered by arbiter007 6 · 0⤊ 0⤋