NAME
MATH 111 - Calculus I
Project 2 - Using Derive
A circular disk of xed radius
r
is used in a humidier. It is partially submerged in water and rotates on
its center. The goal is to determine the height above the water,
h
, at which to x the center so that the
exposed wet surface area is maximized.
1. Find the surface area that is dry, i.e. never is submerged as the disk rotates, as a function of
r
and/or
h
.
2. Find the total surface area that is wet as a function of
r
and/or
h
. (i.e. either submerged or exposed)
3. Find the area of the sector of the circle that is created by radii that intersect the water as function of
r
and/or
h
. (HINT:
Use
as half of the angle from radius to radius of your sector.
)
4. Find the area of the triangle that is in this sector, but above the water, as a function of
r
and/or
h
.
5. Find the surface area that is submerged at any time as a function of
r
and/or
h
.
6. Subtract to nd the desired area, area that is exposed and wet, as a function of
r
and/or
h
.
7. Dene this function in Derive and follow our standard method for nding maxima and minima.
Remember that
r
is the xed radius and
h
is the height, which we can change. (Hint:
Recall
0
h
r
so you know the endpoints and are trying to nd the absolute max on this interval.
)
8. Note your nal answer should be a way to nd
h
as a function of
r
. If Derive will not give an expression
of the form
h
=, with only a function of
r
on the right side, you may need to do some algebra yourself
to nd a nice form.
9. For disks of radii 1, 2, 3, 4, and 5 give the value for
h
that should be used.
