v = pi r^2 h, so
h = v/(pi r^2)
the material used is determined by the surface area
a = 2pi r^2 + 2pi rh
= 2pi r^2 + 2pi r(v/(pi r^2))
= 2pi r^2 + 2v/r
minimum area is when da/dr = 0, so we want
4pi r - 2v/r^2 = 0
2pi r^3 - v = 0
r = ∛(v/(2pi))
See the curve (with v=1) at
http://www.wolframalpha.com/input/?i=2pi+r^2+%2B+2pi+r%281%2F%28pi+r^2%29%29
A drinking cup is made in the shape of a right circular cylinder. for a fixed volume, we wish to make the total material used, the circular bottom and the cylindrical side, as small as possible. Find the ratio of the height to the diameter that minimizes the amount of material used. Hint: Express the height and diameter as a function of the radius r and find the value of r that minimizes the amount of material used.
a) find the equation to maximized or minimized.
b) find the solution
c) showing that your solution is an absolute max or min.
1 answer
3 answers