Variation

-- Section 2.5 --

 

  y varies directly with x:

y = kx 

(k is a constant)

  y varies inversely with x:

    (k is a constant)

  y varies jointly with w and x:

y = kwx   (k is a constant)
  
Find the constant of proportionality.
y is directly proportional to x.  If x = 3, then y = 24.  

Solve

Step

Check

The goal is to find k, the constant.

y = 8x

y = kx Use the formula for direct variation 

If x = 3, then y = 24.  

If x = 3, then y = 24.   Given

y = 8(3)

24 = k(3) Plug in x and y

y = 24

24 = 3k Divide by 3
k = 8 constant of proportionality

y = 8x

Plug 8 in formula for k

Now we have a formula that can be used to find values for y for specific values of x.

Find the constant of proportionality.
y is directly proportional to x.  If x = 12, then y = 4.

[Solution]

Find the constant of proportionality.
L is inversely proportional to the square of d.  If d = 2, then L = 3. 

Solve

Step

Check

The goal is to find k, the constant. Plug in the values 

 
If d = 3, then L = 5

Use the formula for inverse variation 
If d = 3, then L = 5.   Given

 
L = 5

Plug in d and L
Square 3 and multiply both sides by 9
k = 45 constant of proportionality
Plug 45 in formula for k
 

Find the constant of proportionality.
z is inversely proportional to the cube of w.  If w = 2, then z = 3.  

[Solution]

 

R varies jointly with  p and q. If  R = 20, when p = 5 and q = 8.  
Find R , when p = 3 and q = 16

Solve

                                   Step

The goal is to find k, then R.

Plug in the values. Find the constant of proportionality.

R = kpq

Use the formula for joint variation 
with  p and q

If  R = 20, when p = 5 and q = 8

Given

20 = k(5)(8) = 40k

Plug in p, q and R

constant of proportionality

The formula

Plug in 3 and 16 to find R

R = 24

Done!!

H varies jointly with the square g and the square root f. If  H = 160, when g = 4 and g = 25.  
Find H , when g = 1 and f = 81

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada