F.5 variations

1. Express the following variation by an equation.
a varies inversely as (root b) when c is a constant and varies directly as c³ when b is a constant.

a ∝ c³/√b
Hence, a = kc³/√b
where k is proportional constant.


2. a varies directly as the cube of b when c is a constant, and a varies inversely as the square root of c when b is a constant. If a=16 when b=2 and c=9,
(a) write a formula for a in terms of b and c,
(b) find c when a=12 and b=1.

(a)
a ∝ b³/√c
a = kb³/√c
where k is proportional constant.

a = 16 when b = 2 and c = 9:
(16) = k(2)³/√(9)
16 = 8k/3
k = 6

Hence, a = 6b³/√c

(b)
When a = 12 and b = 1:
12 = 6(1)³/√c
√c = 1/2
c = 1/4


3. y varies directly as s and inversely as the cube of t. What will be the percentage change in y if t is doubled and s is tripled?

y ∝ s/t³
y = ks/t³
where k is proportional constant

when s = so and t = to
y = kso/to³

when s = 3so and t = 2to
y = k(3so)/(2t)³
y = (3/8)kso/to³
y = (3/8)yo

Percentage change in y
= {[(3/8)yo - yo]/yo} x 100%
= -62.5%