Page 116 - How To Solve Word Problems In Calculus

EXAMPLE 16
A conical cup is constructed from a circular piece of paper of
radius 5 in by cutting out a sector and joining the resulting
edges. What is the maximum volume of the cup?

Solution
Step1
Join AB to AC

B C r
5
h
A 5

Step2
π 2
The volume of the resulting cone is V = r h.
3
Step3
From the diagram in step 1 it is apparent that
2
2
2
2
h + r = 25. It follows that r = 25 − h .
π
2
V = r h
3
π
2
= (25 − h )h
3
π
3
V(h) = (25h − h ) 0 ≤ h ≤ 5
3
Step4
π
2
V (h) = (25 − 3h )
3
π
2
0 = (25 − 3h )
3
0 = 25 − 3h 2
2
3h = 25
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