Direct Variation

Let x and y denote two quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero number k such that:

The number k is called the constant of proportionality.

Example

A Relation between x and y

x1 2

21

y

Example Cont.

The graph illustrates the relationship between y and x if y varies directly with x and k >0, x ≥0. Also notice that the constant of proportionality is in fact, the slope of the line.

Example

For a certain gas enclosed in a container of fixed volume, the pressure P (in newtons per square meter) varies directly with temperature T in Kelvins. If the pressure is found to be 20 newtons per square meter at a temperature T find a formula that relates pressure P to temperature T. Then find the pressure P when T=120K .

Inverse Variation

Let x and y denote two quantities. Then y varies inversely with x, or y is inversely proportional to x, if there is a nonzero constant k such that:

Example

The Graph illustrates the relationship between y and x if y varies inversely with x and k>0, x>0.

x1 2

21

y

Maximum Weight that can be supported by a piece of pine

  The maximum weight W that can be safely

supported by a 2X4 inches piece of pine varies inversely with its length I. Experiments indicate that the maximum weight that a 10-foot long pine 2X4 can support is 500 pounds. Write a general formula relating the maximum weight W that can be safely supported by a length of 25 feet.

Joint Variation and Combined Variation

When a variable quantity Q is proportional to the product of two or more other variables, we say that Q varies jointly with these quantities. Finally combinations of direct and/or inverse variation may occur. This is usually referred to as combined variation.

Loss of Heat through a Wall

The loss of heat through a wall varies jointly with the area of the wall and the difference between the inside and outside temperatures and varies inversely with the thickness of the wall. Write an equation that relates these quantities.

Force of the Wind on a Window

The force F of the wind on a flat surface positioned at a right angle to the direction of the wind varies jointly with the area, A, of the surface and the square of the speed v of the wind. A wind of 30 miles per hour blowing on a window measuring 4 by 5 feet has a force of 150 pounds. What is the force on a window measuring 3 by 4 feet caused by a wind of 50 miles per hour.