§3.3: Variation
Learning Objectives:
1. Solve Direct Variations
2. Solve Inverse Variations
3. Solve Joint Variations
Direct Variation
Direct Variation
y varies directly with x, or y is directly proportional to x, if there is a nonzero
constant k, such that y = kx, k is called the constant of variation.
Example: A salesperson’s commission varies directly with the number of cars
sold.
Solving Applications of Variation
Example: The distance a cyclist rides over a flat terrain is directly proportional to
the time the cyclist has been riding. A cyclist rode 45 miles in 2.1 hours. How far
did the cyclist travel in 4 hours, assuming no breaks.
Example: Write a variation equation for these statements.
a. The volume of a sphere varies directly with the cube of the radius.
b. The surface area of a sphere varies directly with the square of the radius.
Inverse Variation
Inverse Variation
y varies inversely with x, or y is inversely proportional to x, if there is a nonzero
1
constant k such that y  k   , k is called the constant of variation or
 x
proportionality constant.
Example: The number of days required to paint a room is inversely proportional
to the number of painters.
Example: While driving home to see his family during Spring Break, Jeremy
wonders how much time he could shave off his trip by setting the cruise control a
bit higher. If he behaves and drives the speed limit, which is essentially 70 mph
from door to door, the drive will take 9 hr. Given that travel time varies inversely
with speed,
a. find the constant of variation and write the variation equation. What does the
constant of variation represent and how does this variation equation compare to
the familiar formula distance  rate  time ?
b. Use the formula found in part (a) to estimate the time he could save if he sets
the cruise at 73 mph. Assume he doesn’t lose any time getting pulled over and
ticketed!
c. If he instead decides to ease back and enjoy the scenery, what speed would
he need to maintain to make it home in 10 hr?
Joint Variation
Joint Variations
1. y varies jointly with the product of x and p: y = kxp or
 xp 

 q 
2. y varies jointly with the product of x and p, and inversely with q: y  k 
Example: The body mass index (BMI) is a standard measure of personal health
used by life insurance companies, physical trainers, and weight loss programs. A
person’s BMI varies directly with his weight, but inversely with the square of his
height. If a 5’8” male has a BMI of 24 while weighing 157.8 lb,
a. find the constant of variation and write the variation equation.
b. How much weight would he need to lose to get down to his target BMI of 22?
c. If he instead gains 6.6 lb over the holidays, what is his new BMI?