Cambridge Additional Mathematics

d

d

t

_q

_____

must be measured in

radiansper time unit.

_

AB

C

xcm

20 cm

q

20 cm

30 °

p^403 cm

Applications of differential calculus (Chapter 14) 401

Example 23 Self Tutor

Triangle ABC is right angled at A, and AB=20cm. AbBC increases at a constant rate of 1 ±per

minute. At what rate is BC changing at the instant when AbBC measures 30 ±?

Let AbBC=μ and BC=xcm

Now cosμ=

20

x

=20x¡^1

) ¡sinμ

dμ

dt

=¡ 20 x¡^2

dx

dt

Particular case:

When μ=30±, cos 30±=

20

x

)

p

3

2

=

20

x

) x=p^403

Also, dμ

dt

=1±per min

= 180 ¼ radians per min

Thus ¡sin 30±£ 180 ¼ =¡ 20 £

3

1600

£

dx

dt

) ¡^12 £ 180 ¼ =¡ 803

dx

dt

)

dx

dt

= 360 ¼ £^803 cm per min

¼ 0 : 2327 cm per min

) BC is increasing at approximately 0 : 233 cm per min.

EXERCISE 14F

1 aandbare variables related by the equation ab^3 =40. At the instant when a=5, bis increasing

at 1 unit per second. What is happening toaat this instant?

2 The length of a rectangle is decreasing at 1 cm per minute. However, the area of the rectangle remains

constant at 100 cm^2. At what rate is the breadth increasing at the instant when the rectangle is a square?

3 A stone is thrown into a lake and a circular ripple moves out

at a constant speed of 1 ms¡^1. Find the rate at which the

circle’s area is increasing at the instant when:

a t=2seconds b t=4seconds.

4 Air is pumped into a spherical weather balloon at a constant

rate of 6 ¼m^3 per minute. Find the rate of change in its

surface area at the instant when the radius of the balloon is

2 m.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\401CamAdd_14.cdr Thursday, 10 April 2014 3:56:35 PM BRIAN