basic_engineering_mathematics0

Algebra 45
10. p 2 − 3pq × 2p ÷ 6q + pq
11. (x + 1)(x − 4) ÷ (2x + 2)
12.
1
of 2y + 3y(2y − y)
4
6.5 Direct and inverse proportionality
An expression such as y = 3x contains two variables. For every
value of x there is a corresponding value of y. The variable x is
called the independent variable and y is called the dependent
variable.
When an increase or decrease in an independent variable leads
to an increase or decrease of the same proportion in the dependent
variable this is termed direct proportion. Ify = 3x then y is
directly proportional to x, which may be written as y ∝ x or y = kx,
where k is called the coefficient of proportionality (in this case,
k being equal to 3).
When an increase in an independent variable leads to a
decrease of the same proportion in the dependent variable (or
vice versa) this is termed inverse proportion. Ify is inversely
proportional to x then y ∝ 1 or y = k/x. Alternatively, k = xy,
x
that is, for inverse proportionality the product of the variables is
constant.
Examples of laws involving direct and inverse proportional in
science include:
(i) Hooke’s law, which states that within the elastic limit of
a material, the strain ɛ produced is directly proportional to
the stress, σ , producing it, i.e. ε ∝ σ or ε = kσ .
(ii) Charles’s law, which states that for a given mass of gas at
constant pressure the volume V is directly proportional to
its thermodynamic temperature T, i.e. V ∝ T or V = kT.
(iii) Ohm’s law, which states that the current I flowing through
a fixed resistor is directly proportional to the applied voltage
V , i.e. I ∝ V or I = kV .
(iv) Boyle’s law, which states that for a gas at constant temperature,
the volume V of a fixed mass of gas is inversely
proportional to its absolute pressure p, i.e. p ∝ (1/V )or
p = k/V , i.e. pV = k
Problem 50. If y is directly proportional to x and y = 2.48
when x = 0.4, determine (a) the coefficient of proportionality
and (b) the value of y when x = 0.65
(a) y ∝ x, i.e. y = kx. Ify = 2.48 when x = 0.4, 2.48 = k(0.4)
Hence the coefficient of proportionality,
k = 2.48
0.4 = 6.2
(b) y = kx, hence, when x = 0.65, y = (6.2)(0.65) = 4.03
Problem 51. Hooke’s law states that stress σ is directly
proportional to strain ε within the elastic limit of a
material.When, for mild steel, the stress is 25 × 10 6 pascals,
the strain is 0.000125. Determine (a) the coefficient of proportionality
and (b) the value of strain when the stress is
18 × 10 6 pascals.
(a) σ ∝ ε, i.e. σ = kε, from which k = σ/ε. Hence the
coefficient of proportionality,
k =
25 × 106
0.000125 = 200 × 109 pascals
(The coefficient of proportionality k in this case is called
Young’s Modulus of Elasticity)
(b) Since σ = kε, ε = σ/k
Hence when σ = 18 × 10 6 , strain
ε =
18 × 106
200 × 10 9 = 0.00009
Problem 52. The electrical resistance R of a piece of wire
is inversely proportional to the cross-sectional area A. When
A = 5mm 2 , R = 7.02 ohms. Determine (a) the coefficient
of proportionality and (b) the cross-sectional area when the
resistance is 4 ohms.
(a) R ∝ 1 , i.e. R = k/A or k = RA. Hence, when R = 7.2 and
A
A = 5, the coefficient of proportionality, k = (7.2)(5) = 36
(b) Since k = RA then A = k/R
When R = 4, the cross sectional area, A = 36
4 = 9mm2
Problem 53. Boyle’s law states that at constant temperature,
the volume V of a fixed mass of gas is inversely
proportional to its absolute pressure p. If a gas occupies
a volume of 0.08 m 3 at a pressure of 1.5 × 10 6 pascals
determine (a) the coefficient of proportionality and (b) the
volume if the pressure is changed to 4 × 10 6 pascals..
(a) V ∝ 1 , i.e. V = k/p or k = pV
p
Hence the coefficient of proportionality,
(b) Volume V = k p
k = (1.5 × 10 6 )(0.08) = 0.12 × 10 6
=
0.12 × 106
4 × 10 6 = 0.03 m 3