Direct variation Most variation questions follow a standard form. You are told how 2 (or more) quantities are related to each other. You are given some initial conditions. You are asked to find the value of one of the quantities when the other quantity has a given value. To solve a variation question follow the following steps. Write a proportion statement. Form an equation by adding the constant of variation, k. Sub in the initial values. Solve the equation for k. Rewrite the equation with the new value of k. Answer the question that was asked.

Types of direct variation Types of direct variation include: Linear: Quadratic: Cubic: Square root: With all types of direct variation as x increases, y increases and when x decreases, y decreases. Sometimes the word variation is replaced by “proportion”. Eg: y is proportional to x y is proportional to the square x y is proportional to the cube x y is proportional to the square root x y varies as x y  x y varies as the square of x y  x2 y varies as the cube of x y  x3 y varies as the square root of x y  x

Example 1 The number of metres, m, a ball falls after being dropped off a cliff is directly proportional to the number of seconds, t, it has been falling. The ball falls 45 metres in the first 3 seconds: How far does the ball fall in the first 2.5 seconds? How long will the ball take to fall 125 metres?  write a proportion statement a) m  t2 b) m = 5t2 125 = 5t2 t2 = 125  5 t2 = 25 t = 5 sec.  form an equation, add the constant of variation k m = kt2  sub in the initial values 45 = k × 32  solve for k k = 45  9 = 5 What does this represent?  rewrite the equation m = 5t2  answer the question m = 5 × 2.52 = 31.25 m

Example 2 The distance you can see out to sea in proportional to the square root of your height above sea level. At a height of 25m, you can see 18 nautical miles out to sea. a) How far can you see from 56m above the sea? b) If you can see 50 nautical miles out to sea, how high are you?  write a proportion statement a) b) 50 = 3·6 × h h = 50  3·6 h = 13·89 h = (13·89)2 h = 192·9m  form an equation, add the constant of variation k  sub in the initial values 18 = k × 25  solve for k k = 18  5 = 3·6  rewrite the equation  answer the question d = 3·6 × 56 = 26·9 M

Today’s work Exercise 12F Page 382 Q1, 2, 7 to 12