3.7: Modeling Using Variation

Direct Variation Let x and y denote two quantities. y varies directly with x, or y is directly proportional to x, if there is a nonzero number k such that y = k x. The number k is called the constant of proportionality.

Example The monthly payment, p, on Mr. Cawelti’s student loan varies directly with the amount borrowed, B. If the monthly payment is $20 for every $1000 borrowed, find an equation that relates the monthly payment p to the amount borrowed B. Find the monthly payment p when the amount borrowed B is $80,000.

The volume of tears cried is directly proportional to the amount of time Mr. Muzny spends grading his algebra 1 tests. A 60 minute grading session causes him to cry 1.5 oz. of tears. How much time did he spend grading if he cried 11 oz. of tears?

Inverse Variation

The length of a violin string varies inversely as the frequency of its vibrations. If a string 8 inches long vibrates at a frequency of 640 cycles per second, what is the frequency of a string that is 10 inches long?

While traveling at a constant speed in a car, the centrifugal accelerations passengers feel while the car is turning is inversely proportional to the radius of the turn. If the passengers feel an acceleration of 12 ft/sec 2 when the radius of the turn is 40 feet, find the acceleration the passengers feel when the radius of the turn is 160 feet?

Joint Variation

The time (in hours) it takes a satellite to complete an orbit around the earth varies directly with the radius of the orbit (from the center of the earth) and inversely with the orbital velocity. A satellite completes an orbit 810 miles above the earth in 16 hours at a velocity of 38,000 mph. Solve for the constant of proportionality, and write an equation(Use 3960 as the radius of the earth).