Maths Made Easy Forums Standard Linear Relationships Direct Variation

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Keymaster
Post count: 802

The rate at which water comes out of a house is directly proportional to the square of the hose’s diameter.
a.   If a hose of diameter 3cm gives out water at the rate of 3.6 litres/ second find the rate at which water comes out of a fireman’s hose which is 8cm across.
b.   What diameter hose do I use if I want water to come out at 1 litre/ second?

a. w α d2
w = kd²
3.6 = k × 32
k = 0.4

w = 0.4d2

if d = 8
w = 0.4 × 82
w = 25.6L/s

b. 1 = 0.4d2
2.5 = d2
d = 1.58cm

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Keymaster
Post count: 802

The weight of a sphere is directly proportional to the cube of its radius. If a sphere with radius 4cm has a weight of 5 kilograms find the weight of a sphere of radius 8cm.

w = $$\alpha r^3$$
w = kr3
5 = k × 43
$$k=\frac{5}{64}$$
w = 5/64 × 83
w = 40kg

Keymaster
Post count: 802

If c varies with the cube of v, and when v = 60, c = 12, find the value of c when v = 90.

c  $$\alpha$$ v3
c = kv3
12 = k × 603
12 = k × 216000
k = 12/216000
k = 1/18000
c = 1/18000 × 903
c = 40.5

Keymaster
Post count: 802

The energy e produced by a falling object travelling at speed s is directly proportional to the square of its speed. The energy produced by a certain object falling at a speed of 4m/s is 136 joules. Find the energy produced when the speed of the object is 6.5m/s.

e = ks2
136 = k × 42
k = 136/16
k = 8.5

e = 8.5 × 6.52
e = 359.125

359.125 joules