Cambridge International AS and A Level Mathematics Pure Mathematics 1

Exercise

(^) 5G
P1^
5
10  A fish tank with a square base and no top is to be made from a thin sheet of
toughened glass. The dimensions are as shown.
(i) Write down an expression for the volume V in terms of x and y.
(ii) Write down an expression for the total surface area A in terms of x and y.
The tank needs a capacity of 0.5 m^3 and the manufacturer wishes to use the
minimum possible amount of glass.
(iii) Deduce an expression for A in terms of x only.
(iv) Find ddAx and d
d
2
2

A

x

.

(v) Find the values of x and y that use the smallest amount of glass and

confirm that these give the minimum value.

11  A closed rectangular box is made of thin card, and its length is three times its
width. The height is h cm and the width is x cm.

(i) The volume of the box is 972 cm^3.

Use this to write down an expression for h in terms of x.

(ii) Show that the surface area, A, can be written as A = 6 x^2 +^2592

x

.

(iii) Find d

d

A

x

and use it to find a stationary point.

Find d

d

2

2

A

x

and use it to verify that the stationary point gives the minimum

value of A.

(iv) Hence find the minimum surface area and the corresponding dimensions

of the box.

xP

xP

yP

h

x

3 x