Cambridge Additional Mathematics

hcm

rcm

μ

B C

10 cm

O

lm

xm

36 cm

36 cm

C

D

B

A

y

O x

y=e-x²

Applications of differential calculus (Chapter 14) 397

8 Infinitely many rectangles which sit on thex-axis can

be inscribed under the curve y=e¡x

2

.

Determine the coordinates of C such that rectangle

ABCD has maximum area.

9 Consider the manufacture of cylindrical tin cans of 1 L capacity,

where the cost of the metal used is to be minimised.

a Explain why the heighthis given by h=^1000

¼r^2

cm.

b Show that the total surface areaAis given by

A=2¼r^2 +

2000

r

cm^2.

c Find the dimensions of the can which makeAas small as

possible.

10 A circular piece of tinplate of radius 10 cm has 3 segments

removed as illustrated. The angleμis measured in radians.

a Show that the remaining area is given by

A= 50(μ+ 3 sinμ) cm^2.

b Findμsuch that the areaAis a maximum, and find the

areaAin this case.

11 Sam has sheets of metal which are 36 cm by 36 cm square. He

wants to cut out identical squares which arexcm byxcm from

the corners of each sheet. He will then bend the sheets along the

dashed lines to form an open container.

a Show that the volume of the container is given by

V(x)=x(36¡ 2 x)^2 cm^3.

b What sized squares should be cut out to produce the

container of greatest capacity?

12 An athletics track has two ‘straights’ of lengthlm, and two semicircular ends

of radiusxm. The perimeter of the track is 400 m.

a Show that l= 200¡¼x and write down the possible values thatxmay

have.

b What values oflandxmaximise the shaded rectangle inside the track?

What is this maximum area?

13 A small population of wasps is observed. After t weeks the population is modelled by

P(t)=

50 000

1 + 1000e¡^0 :^5 t

wasps, where 06 t 625.

Find when the wasp population is growing fastest.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\397CamAdd_14.cdr Monday, 7 April 2014 12:29:18 PM BRIAN