Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

5

Exercise

(^) 5G
163
(i) Write down an equation linking x and y.
(ii) Using your answer to part (i), form an expression for the cross-sectional
area, A, in terms of x.
(iii) By considering ddAxandddxA
2
2 , find the dimensions which make the slide as
safe as possible.
5  A carpenter wants to make a box to hold toys. The box is to be made so that its
volume is as large as possible. A rectangular sheet of thin plywood measuring
1.5 m by 1 m is available to cut into pieces as shown.
(i) Write down the dimensions of one of the four rectangular faces in terms of x.
(ii) Form an expression for the volume, V, of the made-up box, in terms of x.
(iii) Find dd andd
d

V

x

V

x

2

2 ..

(iv) Hence find the dimensions of a box with maximum volume, and the

corresponding volume.

6  A piece of wire 16 cm long is cut into two pieces. One piece is 8x cm long and
is bent to form a rectangle measuring 3x cm by x cm. The other piece is bent to
form a square.
(i) Find in terms of x:
(a) the length of a side of the square
(b) the area of the square.
(ii) Show that the combined area of the rectangle and the square is A cm^2
where A = 7 x^2 − 16 x + 16.
(iii) Find the value of x for which A has its minimum value.
(iv) Find the minimum value of A.
[MEI]

cross-section

x cm x cm

y cm

x m

x m

x m

1.5 m

1m

x m

The shaded area is cut

off and not used.