JOINT VARIATION

Variation :

Let y be the sales of a particular product and x be money spent on advertisement of the same product. Clearly x in an independent variable and y is a dependent variable, that is, y is depending on x.

Consider the equation below which relates x and y.

y = 100x + 1000

Here, the variable y is depending on only one variable x.

In the equation above, when x increases, y also increases. This is direct variation and we say this

"y varies directly as x"

Joint Variation :

There are situations where a variable will depend on two or more variables.

For example, consider three variables which take only positive values as follows.

y ----> dependent

x ----> independent

z ----> independent

Consider the equation below which relates x, y and z.

y = 3xz

Here, the variable y is depending on two variables x and z.

In the equation above, when x and z increase, y also increases. This is joint variation and we say this

"y varies jointly as x and z"

"y varies directly as the product of x and z"

Key Concept

If A varies directly as B and C, then

A ∝ BC ----> A = kBC

If A varies inversely as B and C, then

A ∝ 1/(BC) ----> A = k/(BC)

If A varies directly as B and inversely as C, then

A ∝ B/C ----> A = kB/C

where k is the constant of variation (or proportionality).

Example 1 :

Suppose x varies directly as y and z. If y = 3 and z = 4, then x = 24. Find the value of x when y = 7 and z = 4.

Solution :

Since x varies directly as y and z,

x ∝ yz

x = kyz ----(1)

Substitute x = 24, y = 3 and z = 4 to find the value of k.

24 = k(3)(4)

24 = 12k

Divide both sides by 24.

2 = k

Substitute k = 2 in (1).

x = 2yz

Substitute y = 7, z = 4 and evaluate x.

x = 2(7)(4)

x = 56

Example 2 :

Suppose y varies jointly with the square of x and the cube root of z. If x = 5 and z = 8, then y = 25. Find y, if x = 2 and z = 27.

Solution :

Since x varies directly as y and z,

y ∝ (x²)(³√z)

y = k(x²)(³√z) ----(1)

Substitute x = 5, z = 8 and y = 25 to find the value of k.

25 = k(5²)(³√8)

25 = k(25)(2)

25 = 50k

Divide both sides by 50.

0.5 = k

Substitute k = 0.5 in (1).

y = 0.5(x²)(³√z)

Substitute x = 2, z = 27 and evaluate y.

y = 0.5(2²)(³√27)

y = 0.5(4)(3)

y = 6

Example 3 :

Suppose x varies directly with y and inversely with z. If x = 3 and y = 10, then z = 9. Find x when y = 12 and z = 18.

Solution :

Since x varies directly with y and inversely with z,

x ∝ y/z

x = ky/z ----(1)

Substitute x = 3, y = 10 and z = 9 to find the value of k.

3 = k(10)/9

Multiply both sides by 9.

27 = 10k

Divide both sides by 2.7.

2.7 = k

Substitute k = 2.7 in (1).

x = 2.7y/z

Substitute y = 12, z = 18 and evaluate x.

x = 2.7(12)/18

x = 1.8

Example 4 :

The surface area of a cylinder varies jointly as the radius and the sum of the radius and the height. A cylinder with radius 4 cm and height 8 cm has a surface area 96π cm². Find the surface area of a cylinder with radius 3 cm and height 10 cm.

Solution :

Let S represent surface area of the cylinder, r represent radius and h represent height.

Since S varies jointly as r and (r + h),

S ∝ r(r + h)

S = kr(r + h) ----(1)

Substitute S = 96π, r = 4 and h = 8 to find the value of k.

96π = k(4)(4 + 8)

96π = k(4)(12)

96π = 48k

Divide both sides by 48.

2π = k

Substitute k = 2π in (1).

S = 2πr(r + h)

Substitute r = 3, h = 10 and evaluate S.

S = 2π(3)(3 + 10)

S = 2π(3)(13)

Surface area = 78π cm²

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

JOINT VARIATION

Key Concept

Recent Articles

Trigonometry Even and Odd Iidentities

SOHCAHTOA Worksheet

Trigonometry Pythagorean Identities