1.10 Variation

Objective Write mathematical models for direct variation, direct variation as an nth power, inverse variation, and joint variation.

Direct Variation Examine the graph at the right: The linear equation graph at the right shows that as the x value increases, so does the y value increase for the coordinates that lie on this line .    For instance, if x = 2, y = 4. If x = 6 (multiplied by 3), then y = 12 (also multiplied by 3).

where the variable k is called the constant of proportionality. This is a graph of direct variation.  If the value of x is increased, then y increases as well. Both variables change in the same manner.  If x decreases, so does the value of y.  We say that y varies directly as the value of x.  A direct variation between 2 variables, y and x, is a relationship that is expressed as: y = kx where the variable k is called the constant of proportionality.

Direct Variation The following statements are equivalent. 1. y varies directly as x. 2. y is directly proportional to x. 3. y = kx for some nonzero constant k. k is the constant of variation or the constant of proportionality.

Example 1 The simple interest on an investment is directly proportional to the amount of the investment. By investing $2500 in a certain bond issue, you obtained an interest payment of $187.50 at the end of 1 year. Find a mathematical model that gives the interest I for this bond issue at the end of 1 year in terms of the amount invested P.

Direct Variation as an nth Power Another type of direct variation relates one variable to a power of another variable. The area A is directly proportional to the square of the radius r.

Direct Variation as an nth Power The following statements are equivalent. 1. y varies directly as the nth power of x. 2. y is directly proportional to the nth power of x. 3.

Example 2 Neglecting air resistance, the distance s an object falls varies directly as the square of the duration t of the fall. An object falls a distance of 144 feet in 3 seconds. Write an equation relating distance s and duration t. How far will an object fall in 6 seconds?

Inverse Variation (The opposite of direct variation) In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases. For instance, a biker traveling at 8 mph can cover 8 miles in 1 hour.  If the biker's speed decreases to 4 mph, it will take the biker 2 hours (an increase of one hour), to cover the same distance.  Inverse variation:  when one variable increases, the other variable decreases.   

Notice the shape of the graph of inverse variation Notice the shape of the graph of inverse variation. If the value of x is increased, then y decreases.     If x decreases, the y value increases.  We say that y varies inversely as the value of x.  An inverse variation between 2 variables, y and x, is a relationship that is expressed as: y = k / x where the variable k is called the constant of proportionality.

Inverse Variation The following statements are equivalent. 1. y varies inversely as x. 2. y is inversely proportional to x. 3. y = k / x for some constant k.

Example 3 A company has found that the demand for its product varies inversely as the price of the product. When the price is $2.75, the demand is 600 units. Write an equation relating demand d and price p. Approximate the demand when the price is $3.25.

Joint Variation Sometimes more than one variable is involved in a direct variation problem.  In these cases, the problem is referred to as a joint variation.  The formula remains the same, with the additional variables included in the product.  

Joint Variation The following statements are equivalent. 1. z varies jointly as x and y. 2. z is jointly proportional to x and y. 3. z = kxy for some constant k.

Example 4 The maximum load that can be safely supported by a horizontal beam is jointly proportional to the width of the beam and the square of its depth, and inversely proportional to the length of the beam. Determine the change in the maximum safe load und the following conditions. The width of the beam is doubled. The depth of the beam is doubled.

The works of Leonardo Da Vinci display the artist's keen interest in proportions in the human body.  The facial features of the Mona Lisa show numerous applications of the golden section (golden ratio) which Leonardo referred to as the "Divine Proportion" for its esthetically pleasing nature.

As a student of anatomy, Leonardo was fascinated with the mathematical relationships found in the human structure.  His drawing, Vitruvian Man, was a study of the proportions of the human body.  It is said that Leonardo believed the workings of the human body to be an analogy for the workings of the universe.Leonardo made a series of observations concerning the human body.  Among them are statements such as: •  the length of a man's outstretched arms is equal to his height. • the length of the ear is one-third the length of the face.      • the distance from the bottom of the chin to the nose is one-third of the length of the head.      • the length of the hand is one-tenth of a man's height.

QUESTION 1: Leonardo observed that the length of the human face varies directly as the length of the human chin.  If a person whose face length is 9 inches has a chin length of 1.2 inches, what is the length of a person's face whose chin length is 1.5 inches?   QUESTION 2: Leonardo also observed that the human height varies directly as the width of the human shoulders.  If a person 70 inches tall has a shoulder width of 18 inches, what is the height of a person whose shoulder width is 16.5 inches?