Direct & Inverse Variation

EQ: How does a “direct” relationship between two variables compare to an “inverse” relationship between two variables?

Direct Variation

•As x increases, y increases

•As x decreases, y decreases

•When x = 0, y = 0

Inverse Variation

•As x increases, y decreases

•As x decreases, y increases

•Never equals zero

Quick Write – 1 minute

•Think of something in your life that varies directly (increase-increase or decrease-decrease) and describe the relationship

•Think of something in your life that varies inversely (increase-decrease or decrease-increase) and describe that relationship

Direct Inverse

k = “constant of variation”

SOLVE FOR ‘k’

k = “constant of variation”

SOLVE FOR ‘k’

Find the constant of variation (Direct)

•1) 6 = k(5)

•2) 0.5 = k(0.2)

•3) 4 = k(8)

Find the constant of variation (Inverse)•1)

•2)

•3)

Ex 1) If y = -6 when x = -2 find y when x = 4 if…

a. y varies directly with x

  

b. y varies inversely with x

Ex 2) If y = 14 when x = 2, find y when x = 4 if…

a. y varies directly with x

b. y varies inversely with x 

Ex 3) z varies directly with x and inversely with y. When x = 5 and y = 2, z = 10. Find z when x = 4 and y = 8

Problem #1

•The volume V of a gas varies inversely as its pressure P. If V = 80 cubic centimeters when P = 2000 millimeters of mercury, find V when P = 320 millimeters of mercury.

Problem #2

•The length S that a spring will stretch varies directly with the weight F that is attached to the spring. If a spring stretches 20 inches with 25 pounds attached, how far will it stretch with 15 pounds attached

Problem #3•The number of bags of grass seed n

needed to reseed a yard varies directly with the area a to be seeded and inversely with the weight w of a bag of seed. If it takes two 3-lb bags to seed an area of 3,600 ft2, how many 3-lb bags will seed 9000 ft2 ?

Practice

•Check your answers at one of the solution stations

The current in a simple electrical circuit is inversely proportional to the resistance. If the current is 80 amps when the resistance is 50 ohms, find the current when the resistance is 22 ohms.

181.81 amps

Your weight on Mars varies directly with your weight on Earth. A person weighing 125 lbs on Earth weighs 47.25 lbs on Mars, since Mars has less gravity. If you weigh 140 lbs on Earth, how much will you weigh on Mars?

52.92 lbs

The frequency of a vibrating guitar string varies inversely as its length. Suppose a guitar string 0.65 meters long has a frequency of 4.3 per second. What frequency would a string 0.5 meters long have?

5.59 per second

There are about 200 calories in 50 grams of Swiss cheese.  Willie ate 70 grams of this cheese.  About how many calories were in the cheese that he ate if the number of calories varies directly as the weight of the cheese.

280 calories

The number of hours h that it takes m men to assemble x machines varies directly as the number of machines and inversely as the number of men. If four men can assemble 12 machines in four hours, how many men are needed to assemble 36 machines in eight hours?

6 men