# Jointly Proportional

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Jointly proportional is also known as joint variation. It is very much like a direct variation with a difference that the joint proportional involves more than one variable. If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly. In other words we can say that it is jointly proportional to other variables. This can be represented mathematically as y = k * x* z where the variable k is the constant of variation.

Example 1:

The volume of wood in a tree varies jointly as the height and inversely as the square of the girth. The volume of a tree is 144 cubic meters when the height is 20 meters and the girth is 1.5 meters. What will be the height of a tree with a volume of 1000 cubic meters and girth of 2 meters?

Solution:

We set up the equation according to the problem

V = (k * height)/ girth ^2

We plug in the value of V, height and girth to find k

144 = (k * 20)/ 1.5 ^2

k = (144 * 1.5 ^2) / 20 = 16.2

Now we can plug in the new values to find the new height.

1000 = (16.2 * height)/ 2 ^2

Height = (1000 * 4)/ 16.2 = 246.91meters

Example 2:

Given that a varies jointly with b and c. If a = 45 when b= 5 and c = 3 then find the constant of variation?

Solution:

a = k * b * c

45 = k * 5* 3

k = 45 / 15 = 3