Direct variation

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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.

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Types of direct variation
Types of direct variation include Linear Quadr
atic 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
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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
k 45 ? 9 5
? solve for k
What does this represent?
? rewrite the equation
m 5t2
? answer the question
m 5 2.52 31.25 m
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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?
a)
? write a proportion statement
b) 50 36 ?h ?h 50 ? 36 ?h
1389 h (1389)2 h
1929m
? form an equation, add the constant of
variation k
? sub in the initial values
18 k ?25
k 18 ? 5 36
? solve for k
? rewrite the equation
? answer the question
d 36 ?56 269 M
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Todays work
Exercise 12F Page 382 Q1, 2, 7 to 12