Cambridge Additional Mathematics

2 m

xm

piston

μmust be in

so the

dimensions

are correct.

radians

12 cm

A μ

C

B

10 cm

P

Q

R

6 cm

7 cm

μ

y

x

P,(t t)cos sin

t (1 0),

A

piston

xm O

2 m

392 Applications of differential calculus (Chapter 14)

11 In the conversion of sugar solution to alcohol, the chemical reaction obeys the law A=s(1¡e¡kt),

t > 0 wheretis the number of hours after the reaction commences, s is the original sugar

concentration (%), andAis the alcohol produced, in litres.

a FindAwhen t=0.

b Suppose s=10and A=5after 3 hours.

i Findk. ii Find the speed of the reaction at time 5 hours.

c Show that the speed of the reaction is proportional to A¡s.

Example 17 Self Tutor

Find the rate of change in the area of

triangle ABC asμchanges, at the time

when μ=60±.

Area A=^12 £ 10 £ 12 £sinμ fArea=^12 bcsinAg

) A= 60 sinμ cm^2

)

dA

dμ

= 60 cosμ

When μ=¼ 3 , cosμ=^12

)

dA

dμ

=30cm^2 per radian

12 Find the rate of change in the area of triangle PQR asμchanges, at

the time when μ=45±.

13 On the Indonesian coast, the depth of water at time t hours after midnight is given by

d=9:3+6:8 cos(0: 507 t) metres.

a Find the rate of change in the depth of water at 8 : 00 am.

b Is the tide rising or falling at this time?

14 A piston is operated by rod [AP] attached to a

flywheel of radius 1 m. AP=2m. P has

coordinates (cost,sint) and point A is (¡x,0).

a Show that x=

p

4 ¡sin^2 t¡cost.

b Find the rate at whichxis changing at the

instant when:

i t=0 ii t=¼ 2 iii t=^23 ¼

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\392CamAdd_14.cdr Monday, 7 April 2014 11:56:48 AM BRIAN