1. Joint Variation

2. Consider the curved surface area A of the right circular cone. r
is a constant.
A = r
If r is fixed,
say r = 3.5, r
3.5 …  A
When  = 6
When  = 7
When  = 8 6 7 8 … 66 77 88 …
A varies directly as .
is the variation constant.

3. Consider the curved surface area A of the right circular cone. r
is a constant.
A = r
If  is fixed,
say  = 7,
When r = 2
When r = 3
When r = 4 r  7 … A 2 3 4 … 44 66 88 …
A varies directly as r.
 is the variation constant.

4. Consider the curved surface area A of the right circular cone. r
is a constant.
A = r
If neither r nor  is fixed,
When r = 2 and  = 7
When r = 3.5 and  = 8
When r = 7 and  = 5 r  A 2
3.5 7 … 7 8 5 … 44 88
110 …
A varies directly as r .
which is a constant
is the variation constant.

5. If r is fixed
A varies directly as .
If  is fixed
A varies directly as r.
If neither r nor  is fixed
A varies directly as r .
We say that A varies jointly as r and .
In symbols, we write A  r .

6. Joint Variation
When a quantity varies directly as the product of two or more quantities, the relation among these quantities is called a joint variation.
In fact, some joint variations also involve inverse variation.
Moreover, a combination of direct variation and inverse variation is also a joint variation.
For example,
z varies directly as x and inversely as y. kx
, where k is a non-zero constant
z = y
k is called the variation constant.

7. Follow-up question
If E varies jointly as m and h, and E = 200 when
m = 4 and h = 5, find
an equation connecting m, h and E,
the value of E when m = 3 and h = 2.
(a)
∵ E varies jointly as m and h.
∴ E = kmh, where k ¹ 0
k is the variation constant.
By substituting m = 4, h = 5 and E = 200 into the equation, we have
200 = k(4)(5)
k = 10
∴ E = 10mh

8. Follow-up question
If E varies jointly as m and h, and E = 200 when
m = 4 and h = 5, find
an equation connecting m, h and E,
the value of E when m = 3 and h = 2.
(b)
When m = 3 and h = 2,
E = 10(3)(2)
= 60