PROBLEM 4 : A container in the shape of a right circular cylinder

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PROBLEM 4 : A container in the shape of a right circular cylinder with no top
has surface area 3 ft.2 What height h and base radius r will maximize the
volume of the cylinder ?
SOLUTION
Let variable r be the radius of the circular base and variable h the height of the
cylinder.
The total surface area of the cylinder is given to be
(area of base) + (area of the curved side)
,
so that
or
.
We wish to MAXIMIZE the total VOLUME of the cylinder
V = (area of base) (height)
.
However, before we differentiate the right-hand side, we will write it as a
function of r only. Substitute for h getting
.
Now differentiate this equation, getting
=0
for
r=1 or r=-1 .
But
since variable r measures a distance and r > 0 . Since the base of
the box is a circle and there are
ft.2 of material, it follows that
See the adjoining sign chart for V' .
If
r=1 ft. and h=1 ft. ,
then
ft.3
is the largest possible volume of the cylinder.
.
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