Apps. and Interpretation for IBDP Maths Ebook 1 | Page 213

(b)
Find d V
dr . [2]
(c)
Find the radius of the can when the volume reaches its maximum.
(d) (i) Find the maximum volume of the can in terms of � .
[3]
(e)
(ii) Hence, write down the range of V .
Find the values of the radii when the volume of the can is
3
47 cm .
[3]
[3]
15
2. In a Design & Technology lesson, a teacher uses a square paper with length 64 cm to
make an open box. Firstly, four squares with lengths x cm on four corners are removed.
Then the four rectangular flaps are folded up to form an open box, such that the base is a
square.
(a) (i) Write down the length of the square base of the open box.
(b)
3 2
(ii) Show that V � 4x � 256x � 4096x
.
[3]
Find d V
dx . [2]
(c) Find the value of x when the volume reaches its maximum.
[3]
(d) (i) Find the maximum volume of the open box, giving the answer correct to
the nearest
3
cm .
(e)
(ii) Hence, write down the range of V .
[3]
All surfaces of the open box has to be painted afterwards. Find the total surface
area that has to be painted when its volume reaches its maximum.
[3]
3. A box in the shape of triangular prism with height h is produced such that the total
surface area A is
2
100 cm . The cross-sectional area of the box is an equilateral triangle
with side length r . Let V be the volume of the box.
(a) (i) Using
sin 60
�
3
, show that
2
2
200 � 3r
h � .
6r
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