Apps. and Interpretation for IBDP Maths Ebook 1 | Page 211

59
Paper 2 – Optimization Problems involving 3-D Objects
Example
A toy in the shape of circular cone is produced such that the sum of its height h and its
radius r is 45 cm. Let V be the volume of the toy.
(a)
(b)
2 1 3
Show that V �15�r � �r
.
3
[2]
Find d V
dr . [2]
15
(c)
Find the radius of the toy when the volume reaches its maximum.
(d) Find the maximum volume of the toy in terms of � .
(e)
(f)
Find the total surface area of the toy when its volume reaches its maximum.
Find the values of the radii when the volume of the toy is
3
3000� cm .
[3]
[2]
[3]
[3]
Solution
(a) h�r
� 45
h�45� r
M1
1 2
V � � r h
3
1 2
V � � r (45 � r)
3
A1
2 1 3
V �15�r � �r
3
AG N0
[2]
(b)
dV
1 2
�15 �(2 r) � �(3 r )
dr
3
(A1) for correct derivatives
dV
2
�30�r� �r
dr
A1 N2
[2]
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