Variation:
- how one quantity changes (varies) in relation to another quantity
- quantities can vary directly, jointly, indirectly (inversely), or
combined
Direct Variation:
- a dependent variable moves in the same direction as an independent
variable
- described by formulas of the form 𝑦 = 𝑘𝑥 𝑛
o when 𝑥 increases, 𝑦 increases
o when 𝑥 decreases, 𝑦 decreases
- 𝑘 is known as the constant of variation or the constant of
proportionality
o 𝑘 can assume the units of any variables; this means we do not
need to make units consistent when working with variation
problems
- Examples:
o the circumference of a circle varies directly as the radius
 𝐶 = 2𝜋𝑟;
the constant of variation 𝑘 = 2𝜋
o the area of a circle varies directly as the square of the radius
 𝐴 = 𝜋𝑟 2 ;
the constant of proportionality 𝑘 = 𝜋
Example 1: Express the following statement as a formula that involves
the given variables and a constant of proportionality 𝑘, and then determine
the value of 𝑘 from the given conditions.
𝑏 is directly proportional to 𝑡. If 𝑡 = 13, then 𝑏 = 7.
Example 2: Express the following statement as a formula that involves
the given variables and a constant of proportionality 𝑘, and then determine
the value of 𝑘 from the given conditions.
1
𝑥 is directly proportional to 𝑦 2 . If 𝑦 = 3, then 𝑥 = 𝜋.
Example 3: Express the following statement as a formula that involves
the given variables and a constant of proportionality 𝑘, and then determine
the value of 𝑘 from the given conditions.
𝑠 varies directly as the fourth root of 𝑡. If 𝑡 = 16, then 𝑠 = 18.
Example 4: An objects weight on the moon 𝑀 varies directly as its weight
on Earth 𝐸.
a. Express the statement above as a formula, including the constant of
variation 𝑘.
b. Neil Armstrong weighed 360 pounds on Earth (with all of his
equipment) and 60 pounds on the moon. Use this information to
determine the value of 𝑘.
c. What is the moon weight of a person who weighs 186 pounds on
Earth?
Earth Moon
Weight Weight
0
0
60
10
120
20
180
30
240
40
300
50
360
60
60
50
40
𝑀
30
20
10
0
0
60
120
180
240
𝐸
300
360
Example 5: The volume of a sphere 𝑉 varies directly with the cube of the
radius 𝑟.
a. Express the statement above as a formula, including the constant of
variation 𝑘.
b. A sphere with a radius of 3 will have a volume of 36𝜋. Use this
information to determine the value of 𝑘.
c. What is the volume of a sphere with a radius of 2?
Answers to Examples:
7
1. 𝑏 = 𝑘𝑡 ; 𝑘 = 13 ; 2.
4
𝑥 = 𝑘𝑦 2 ; 𝑘 = 9𝜋 ; 3. 𝑠 = 𝑘 √𝑡 ; 𝑘 = 9 ;
1
4a. 𝑀 = 𝑘 ∙ 𝐸 ; 4b. 𝑘 = 6 ; 4c. 𝑀 = 31 𝑝𝑜𝑢𝑛𝑑𝑠 ;
5a. 𝑉 = 𝑘𝑟 3 ; 5b. 𝑘 =
4𝜋
3
; 5c. 𝑉 =
32𝜋
3
;