You already know relationships where one variable varies directly or inversely with another variable. Now you will look at relationships where one variable varies directly with two or more other variables but does not vary inversely with any other variable.

1. MATHEMATICS 9
Prepared by: Beverly Gelbolingo

2. joint VARIATION
◂ Let x, y and z denote three quantities. y varies
jointly as x and z if there is some positive
constant k such that:
𝑦 = 𝑘𝑥𝑧
◂ The number k is called the constant of variation.

3. Solution:
𝑦 = 𝑘𝑥𝑧
45 = 𝑘 18 10
45 = 180𝑘
45
180
= 𝑘
1
4
= 𝑘
1
𝑦 =
1
4
𝑥𝑧
2
𝑦 =
1
4
(20)(30)
𝑦 = 150
3
Example 1:
◂ Find an equation of
variation in which y
varies jointly as x
and z and 𝑦 = 45
when 𝑥 = 18 and
𝑧 = 10 . Find y
when 𝑥 = 20 and
𝑧 = 30.
joint VARIATION
Thus, y=150 when x=20 and z=30.

4. TRY THIS!!!
◂ Find an equation of variation in which p varies
jointly as q and t and p = 60 when 𝑞 = 24
and 𝑡 = 5. Find p when q = 12 and t = 4.
joint VARIATION

5. Solution:
Understand the problem.
Let:
V (y) = the volume of a right
circular cone
r (x) = the radius of the base
h (z) = the height
V varies jointly as r and h.
1
Example 2:
◂ The volume of a right circular cone
varies jointly with the square of its
radius and its height. An ice cream
cone 12 cm high with a radius of 3 cm
has a volume of 113.04𝑐𝑚3. If the
radius of this cone is decreased by
1
2
𝑐𝑚, by how much will its volume
decrease?
joint VARIATION

6. Solution:
Understand the
problem.
Let:
V (y) = the volume
of a right circular
cone
r (x) = the radius of
the base
h (z) = the height
V varies jointly as
r and h.
1
Example 2:
◂ The volume of a right
circular cone varies jointly
with the square of its radius
and its height. An ice
cream cone 12 cm high
with a radius of 3 cm has a
volume of 113.04𝑐𝑚3
. If
the radius of this cone is
decreased by
1
2
𝑐𝑚, by
how much will its volume
decrease?
joint VARIATION
Solution:
Write the equation
𝑉 = 𝑘𝑟2
ℎ
113.04 = 𝑘(3)2
12
113.04 = 𝑘 9 12
113.04 = 𝑘 108
113.04
108
=
𝑘(108)
108
1.047 = 𝑘
𝑽 = 𝟏. 𝟎𝟒𝟕𝒓 𝟐
𝒉
2

7. Solution:
Understand the
problem.
Let:
V (y) = the volume
of a right circular
cone
r (x) = the radius of
the base
h (z) = the height
V varies jointly as
r and h.
1
Example 2:
◂ The volume of a right
circular cone varies jointly
with the square of its radius
and its height. An ice
cream cone 12 cm high
with a radius of 3 cm has a
volume of 113.04𝑐𝑚3
. If
the radius of this cone is
decreased by
1
2
𝑐𝑚, by
how much will its volume
decrease?
joint VARIATION
Solution:
Write the equation
𝑉 = 𝑘𝑟2
ℎ
113.04 = 𝑘(3)2
12
113.04 = 𝑘 9 12
113.04 = 𝑘 108
113.04
108
=
𝑘(108)
108
1.047 = 𝑘
𝑽 = 𝟏. 𝟎𝟒𝟕𝒓 𝟐 𝒉
2
Solution:
Solve the equation
𝑽 = 𝟏. 𝟎𝟒𝟕𝒓 𝟐
𝒉
𝑽 = 𝟏. 𝟎𝟒𝟕 𝟐. 𝟓 𝟐
𝟏𝟐
𝑽 = 𝟕𝟖. 𝟓𝟐𝟓𝒄𝒎 𝟑
3

8. Solution:
Understand the
problem.
Let:
V (y) = the volume of a
right circular cone
r (x) = the radius of the
base
h (z) = the height
V varies jointly as r
and h.
1
Example 2:
◂ The volume of a right
circular cone varies jointly
with the square of its radius
and its height. An ice
cream cone 12 cm high
with a radius of 3 cm has a
volume of 113.04𝑐𝑚3
. If
the radius of this cone is
decreased by
1
2
𝑐𝑚, by
how much will its volume
decrease?
joint VARIATION Solution:
Write the equation
𝑉 = 𝑘𝑟2
ℎ
113.04 = 𝑘(3)212
113.04 = 𝑘 9 12
113.04 = 𝑘 108
113.04
108
=
𝑘(108)
108
1.047 = 𝑘
𝑽 = 𝟏. 𝟎𝟒𝟕𝒓 𝟐
𝒉
2
Solution:
Solve the equation
𝑽 = 𝟏. 𝟎𝟒𝟕𝒓 𝟐
𝒉
𝑽 = 𝟏. 𝟎𝟒𝟕 𝟐. 𝟓 𝟐
𝟏𝟐
𝑽 = 𝟕𝟖. 𝟓𝟐𝟓𝒄𝒎 𝟑
3 The volume of the cone is
decreased by 𝟕𝟖. 𝟓𝟐𝟓𝒄𝒎 𝟑.
(since 𝟏𝟏𝟑. 𝟎𝟒𝒄𝒎 𝟑
−
𝟕𝟖. 𝟓𝟐𝟓𝒄𝒎 𝟑 =
𝟑𝟒. 𝟓𝟏𝟓𝒄𝒎 𝟑
)

9. TRY THIS!!!
◂ The area of a triangle varies jointly with its base
and height. If 𝑏 = 16 𝑐𝑚 and ℎ = 13 𝑐𝑚,
then the area is 104𝑐𝑚2
.
a. What is the constant of variation?
b. What is the area when b = 9 𝑐𝑚 and
ℎ = 12 𝑐𝑚?
joint VARIATION

10. COMBINED VARIATION
◂ Let x, y and z denote three quantities. y varies
directly as x and inversely as z if there is some
positive constant k such that:
𝑦 =
𝑘𝑥
𝑧
◂ The number k is called the constant of variation.

11. Solution:
𝑦 =
𝑘𝑥
𝑧
4 =
𝑘(6)
3
3 4 =
𝑘(6)
3
3
3(4) = 𝑘 6
12 = 𝑘(6)
12
6
=
𝑘(6)
6
2 = 𝑘
1 𝑦 =
2𝑥
𝑧
2
𝑦 =
2(15)
10
𝑦 =
30
10
𝑦 = 3
3
Example 1:
◂ Find an equation of
combined variation in
which y varies directly as
x and inversely as z.
One set of values is
𝑦 = 4 when 𝑥 = 6
and 𝑧 = 3 . Find y
when 𝑥 = 15 and
𝑧 = 10.
COMBINED VARIATION
Thus, y=3 when
x=15 and z=10.