Proportion Direct proportion Y is directly proportional to x Y is proportional to x Y varies as x Y  x Y = k x

Direct proportion Y is directly proportional to the square of x Y is proportional to the square of x Y varies as the square of x Y  x2 Y = k x2

1 Direct proportion Y is directly proportional to the square root of x Y is proportional to the square root of x Y varies as the square root of x Y  Y = k

2 Given that y is directly proportional to the square of x and that y = 12 when x = 2. Find (i) y when x = 9 (ii) x when y = 108

3 Given that y is directly proportional to the square root of x and that y = 20 when x = 25. Find (i) y when x = 81 (ii) x when y = 144

4 Example The pressure of water P varies as the depth, d, below the surface. If the pressure is 180 N/cm3 at a depth of 3m, calculate the pressure at a depth of 7m the depth at which the pressure would exceed 1500 N/cm3

5 Example The resistance, R to the motion of a car is proportional to the square of its speed, v. If the resistance is 800 N at a speed of 20 m/s what is the resistance at a speed of 12 m/s? At what speed would the car be travelling if the resistance is 1500 N?

6 Example When an object falls from rest, the distance y which it falls is proportional to the square of the time, t taken in falling. If y = 32 when t = 4, find the value of y when t = 5.

7 Example When a car goes round a corner, its maximum speed v is proportional to the square root of the radius r of the corner. If v = 20 when r = 100, find v when r = 25.

8 Example The value V of a diamond is proportional to the square of its weight W. If a diamond weighing 10 grams is worth £200, find a) the value of a diamond weighing 30 grams b) the weight of a diamond worth £5000.

9 Example The energy E stored in an elastic band varies as the square of its extension. When the elastic band is extended by 3cm, the energy stored is 243 Joules, What is the energy stored when the extension is 5cm? What is the extension when the stored energy is 36 Joules?