Cambridge Additional Mathematics

10 cm

μμ

end view

q°^10 cm join

A^10 cm C

B

sector

®

3 m

4 m

μ

2 km

Q

PR

398 Applications of differential calculus (Chapter 14)

14 When a new pain killing injection is administered, the effect is modelled by E(t) = 750te¡^1 :^5 t units,

where t> 0 is the time in hours after the injection.

At what time is the drug most effective?

15 A symmetrical gutter is made from a sheet of metal 30 cm wide

by bending it twice as shown.

a Deduce that the cross-sectional area of the gutter is given by

A= 100 cosμ(1 + sinμ).

b Show that

dA

dμ

=0when sinμ=^12 or¡ 1.

c For what value ofμdoes the gutter have maximum carrying

capacity? Find the cross-sectional area for this value ofμ.

16 A sector of radius 10 cm and angleμ±is bent to form a conical cup as shown.

becomes

when edges [AB] and [CB]

are joined with tape.

Suppose the resulting cone has base radiusrcm and heighthcm.

a Show using the sector that arc AC=

μ¼

18

.

b Explain why r=

μ

36

. c Show that h=

q

100 ¡

¡μ

36

¢ 2

.

d Find the cone’s capacityV in terms ofμonly.

e Findμwhen V(μ) is a maximum.

17 Hieu can row a boat across a circular lake of radius 2 km

at 3 km h¡^1. He can walk around the edge of the lake at

5 km h¡^1.

What is the longest possible time Hieu could take to get

from P to R by rowing from P to Q and then walking from

QtoR?

18 In a hospital, two corridors 4 m wide and 3 m wide meet

at right angles. What is the maximum possible length of an

X-ray screen which can be carried upright around the corner?

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\398CamAdd_14.cdr Monday, 7 April 2014 2:39:31 PM BRIAN