Page 106 - How To Solve Word Problems In Calculus

Step2
The volume of a right circular cylinder of radius r and
2
height h is V = πr h.
Step3
If we draw a radius to the point where the sphere and
cylinder meet, we see from the theorem of Pythagoras that

2
h
2 2 Since the volume equation re-
r + = 5 quires r , it is more convenient
2
2
to solve for r in terms of h. Al-
h 2 though solving for h in terms of
2
r + = 25 r would also work, an extra rad-
4 ical is introduced, which makes
h 2 the calculus more difficult.
2
r = 25 −
4
Substituting into the volume equation obtained in step 2,

2
h
V(h) = π 25 − h
4
0 ≤ h ≤ 10

1 3
= π 25h − h
4
Step4
3
2
V (h) = π 25 − h
4

3 2
0 = π 25 − h
4
3 2
0 = 25 − h
4
3 2
h = 25
4
100
2
h =
3
10
h = √
3
93