Power of Maths Paper 1 OL

Power of Maths: Paper 1 – Section 8 (d) Motion EXAMPLE 28

A body moves in a straight line for 10 seconds, its distance s, in metres, from a fi xed point (s = 0) is given by s = 25 t 2 − 5 seconds.

_ 3 t 3 , 0 ≤ t ≤ 10 where t is in

Find its maximum speed and the distance it is from its starting point (t = 0) when its speed is at a maximum.

Solution 1. v (speed) 5. s = 25 t 2 − 5

v = ds __

6. dv ___

_ 3 t 3

dt = 50t − 5 t 2

dt = 50 − 10t 50 − 10t = 0

t = 5 [There is only one stationary point.]

7. v max = 50(5) − 5(25) = 125 m/s s = 25(5 ) 2 − 5

_ 3 (5 ) 3 = 416 2

EXERCISE 8

1. A 20 m length of cable is bent into a rectangular ring circuit, as shown.

20 m h x If the length of the rectangle is x:

(a) express the breadth h of the rectangle in terms of x,

(b) express the area of the rectangle in terms of x,

(c) f ind the area of the rectangle with maximum area that can be constructed in this way.

2. A rectangular sheet of plastic 20 cm by 200 cm is used to make a gutter.

x 200 cm 20 cm x cm h x cm 200 cm

The gutter is made by bending the sheet of plastic along the dotted lines of width x cm.

332 x 2 cm (a) Express, in terms of x:

(i) the width h of the gutter, (ii) the volume V of the gutter.

(b) Find x, to maximize the volume of the gutter.

3. Artwork with a perimeter of 128 cm is mounted inside a rectangular frame. A 1 cm border at the top and the bottom and a border of 2 cm at each side surrounds the artwork, as shown.

1 cm h Artwork x 1 cm

(a) If x is the length of the artwork, express the width h of the artwork in terms of x.

(b) Find an expression for the area A of the frame in terms of x.

(c) Find the maximum possible area of the frame.

4. A box (cuboid) has a square base of side x cm. If the sum of the length of one of the square bases and the height h is 27 cm:

(a) express h in terms of x, (b) express the volume V of the box in terms of x,

(c) fi nd the maximum possible volume of such a box.

2 cm

_ 3 m

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