see
http://www.jiskha.com/display.cgi?id=1395265529
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.
I found the answer to be height/diameter=1/2 when I searched online but there was no work shown. I need someone to show me works. thanks!
3 answers
vol = pi r^2 h
so
h = V/(pi r^2)
area = pi r^2 + 2 pi r h
area = A = pi r^2 + 2 pi r V/(pi r^2)
A = pi r^2 + 2 V /r
dA/dr = 0 = 2 pi r -2 V /r^2
V / r^2 = pi r
V = pi r^3
but
V = h (pi r^2)
so
h (pi r^2) = pi r^3
h = r
or
h = D/2
so
h = V/(pi r^2)
area = pi r^2 + 2 pi r h
area = A = pi r^2 + 2 pi r V/(pi r^2)
A = pi r^2 + 2 V /r
dA/dr = 0 = 2 pi r -2 V /r^2
V / r^2 = pi r
V = pi r^3
but
V = h (pi r^2)
so
h (pi r^2) = pi r^3
h = r
or
h = D/2
LOL - I do not even remember doing that !