Cambridge Additional Mathematics

ym

xm

open

end view

24 cm

y

x

C A

B

y=e-2x

O

A

B C

D x

y

y=k-x 2

O

404 Applications of differential calculus (Chapter 14)

13 Rectangle ABCD is inscribed within the parabola

y=k¡x^2 and thex-axis, as shown.

a If OD=x, show that the rectangle ABCD has area

function A(x)=2kx¡ 2 x^3.

b If the area of ABCD is a maximum when

AD=2

p

3 , findk.

14 A particle moves in a straight line along thex-axis with position given by

x(t) = 3 + sin(2t) cm aftertseconds.

a Find the initial position, velocity, and acceleration of the particle.

b Find the times when the particle changes direction during 06 t 6 ¼ seconds.

c Find the total distance travelled by the particle in the first¼seconds.

15 A rectangular gutter is formed by bending a 24 cm wide

sheet of metal as shown.

Where must the bends be made in order to maximise the

capacity of the gutter?

16 A manufacturer of open steel boxes has to make one with a

square base and a capacity of 1 m^3. The steel costs $ 2 per

square metre.

a If the base measuresxmbyxm and the height isym,

findyin terms ofx.

b Hence, show that the total cost of the steel is C(x)=2x^2 +^8

x

dollars.

c Find the dimensions of the box which would cost the least in steel to make.

17 A particle P moves in a straight line with position from O given by s(t)=15t¡

60

(t+1)^2

cm,

wheretis the time in seconds, t> 0.

a Find velocity and acceleration functions for P’s motion.

b Describe the motion of P at t=3seconds.

c For what values oftis the particle’s speed increasing?

18 Infinitely many rectangles can be inscribed under the curve

y=e¡^2 x as shown.

Determine the coordinates of A such that the rectangle

OBAC has maximum area.

19 A man on a jetty pulls a boat directly towards him so the rope is coming in at a rate of 20 metres

per minute. The rope is attached to the boat 1 m above water level, and the man’s hands are 6 m

above the water level. How fast is the boat approaching the jetty at the instant when it is 15 m

from the jetty?

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\404CamAdd_14.cdr Monday, 7 April 2014 2:51:10 PM BRIAN