Solving simple equations 79
Now try the following Practice Exercise
Practice Exercise 44 Practical problems
involving simple equations (answers on
page 344)
1. A formula used for calculating resistance of a
cable is R =
ρ L
a
. Given R = 1.25, L = 2500
and a = 2 × 10 −4 , find the value of ρ.
2. Force F newtons is given by F = ma, where
m is the mass in kilograms and a is the acceleration in metres per second squared. Find the
acceleration when a force of 4 kN is applied
to a mass of 500 kg.
3. PV = m RT is the characteristic gas equation. Find the value of m when
P = 100 × 10 3 , V = 3.00, R = 288 and
T = 300.
4. When three resistors R 1 , R 2 and R 3 are connected in parallel, the total resistance R T is
determined from
1
R T
=
1
R 1
+
1
R 2
+
1
R 3
(a) Find the total resistance when
R 1 = 3 , R 2 = 6 and R 3 = 18 .
(b) Find the value of R 3 given that
R T = 3 , R 1 = 5 and R 2 = 10 .
5. Six digital camera batteries and 3 camcorder
batteries cost £96. If a camcorder battery
costs £5 more than a digital camera battery,
find the cost of each.
6. Ohm’s law may be represented by I = V /R,
where I is the current in amperes, V is
the voltage in volts and R is the resistance
in ohms. A soldering iron takes a current
of 0.30 A from a 240 V supply. Find the
resistance of the element.
7. The distance, s, travelled in time t seconds is
given by the formula s = ut +
1
2
a t 2 where
u is the initial velocity in m/s and a is the
acceleration in m/s 2 . Calculate the acceleration of the body if it travels 165 m in 3 s, with
an initial velocity of 10 m/s.
Here are some further worked examples on solving
simple equations in practical situations.
Problem 24. The extension x m of an aluminium
tie bar of length l m and cross-sectional area A m 2
when carrying a load of F Newtons is given by the
modulus of elasticity E = Fl/ Ax. Find the
extension of the tie bar (in mm) if
E = 70 × 10 9 N/m 2 , F = 20 × 10 6 N, A = 0.1m 2
and l = 1.4 m
E = Fl/ Ax, hence
70 × 10
9 N
m 2 =
(20 × 10 6 N)(1.4 m)
(0.1m 2 )(x)
(the unit of x is thus metres)
70 × 10
9
× 0.1 × x = 20 × 10
6
× 1.4
x =
20 × 10 6 × 1.4
70 × 10 9 × 0.1
Cancelling gives
x =
2 × 1.4
7 × 100
m
=
2 × 1.4
7 ×100
× 1000 mm
= 4 mm
Hence, the extension of the tie bar, x = 4 mm.
Problem 25. Power in a d.c. circuit is given by
P =
V 2
R
where V is the supply voltage and R is the
circuit resistance. Find the supply voltage if the
circuit resistance is 1.25 and the power measured
is 320 W
Since P =
V 2
R
, then 320 =
V 2
1.25
(320)(1.25) = V
2
i.e.
V
2
= 400
Supply voltage,
V =
√
400 = ±20 V
Problem 26. A painter is paid £6.30 per hour for a
basic 36 hour week and overtime is paid at one and
a third times this rate. Determine how many hours
the painter has to work in a week to earn £319.20
Basic rate per hour = £6.30 and overtime rate per hour
= 1
1
3
× £6.30 = £8.40
Let the number of overtime hours worked = x
Then,
(36)(6.30) + (x)(8.40) = 319.20
226.80 + 8.40x = 319.20
Now try the following Practice Exercise
Practice Exercise 44 Practical problems
involving simple equations (answers on
page 344)
1. A formula used for calculating resistance of a
cable is R =
ρ L
a
. Given R = 1.25, L = 2500
and a = 2 × 10 −4 , find the value of ρ.
2. Force F newtons is given by F = ma, where
m is the mass in kilograms and a is the acceleration in metres per second squared. Find the
acceleration when a force of 4 kN is applied
to a mass of 500 kg.
3. PV = m RT is the characteristic gas equation. Find the value of m when
P = 100 × 10 3 , V = 3.00, R = 288 and
T = 300.
4. When three resistors R 1 , R 2 and R 3 are connected in parallel, the total resistance R T is
determined from
1
R T
=
1
R 1
+
1
R 2
+
1
R 3
(a) Find the total resistance when
R 1 = 3 , R 2 = 6 and R 3 = 18 .
(b) Find the value of R 3 given that
R T = 3 , R 1 = 5 and R 2 = 10 .
5. Six digital camera batteries and 3 camcorder
batteries cost £96. If a camcorder battery
costs £5 more than a digital camera battery,
find the cost of each.
6. Ohm’s law may be represented by I = V /R,
where I is the current in amperes, V is
the voltage in volts and R is the resistance
in ohms. A soldering iron takes a current
of 0.30 A from a 240 V supply. Find the
resistance of the element.
7. The distance, s, travelled in time t seconds is
given by the formula s = ut +
1
2
a t 2 where
u is the initial velocity in m/s and a is the
acceleration in m/s 2 . Calculate the acceleration of the body if it travels 165 m in 3 s, with
an initial velocity of 10 m/s.
Here are some further worked examples on solving
simple equations in practical situations.
Problem 24. The extension x m of an aluminium
tie bar of length l m and cross-sectional area A m 2
when carrying a load of F Newtons is given by the
modulus of elasticity E = Fl/ Ax. Find the
extension of the tie bar (in mm) if
E = 70 × 10 9 N/m 2 , F = 20 × 10 6 N, A = 0.1m 2
and l = 1.4 m
E = Fl/ Ax, hence
70 × 10
9 N
m 2 =
(20 × 10 6 N)(1.4 m)
(0.1m 2 )(x)
(the unit of x is thus metres)
70 × 10
9
× 0.1 × x = 20 × 10
6
× 1.4
x =
20 × 10 6 × 1.4
70 × 10 9 × 0.1
Cancelling gives
x =
2 × 1.4
7 × 100
m
=
2 × 1.4
7 ×100
× 1000 mm
= 4 mm
Hence, the extension of the tie bar, x = 4 mm.
Problem 25. Power in a d.c. circuit is given by
P =
V 2
R
where V is the supply voltage and R is the
circuit resistance. Find the supply voltage if the
circuit resistance is 1.25 and the power measured
is 320 W
Since P =
V 2
R
, then 320 =
V 2
1.25
(320)(1.25) = V
2
i.e.
V
2
= 400
Supply voltage,
V =
√
400 = ±20 V
Problem 26. A painter is paid £6.30 per hour for a
basic 36 hour week and overtime is paid at one and
a third times this rate. Determine how many hours
the painter has to work in a week to earn £319.20
Basic rate per hour = £6.30 and overtime rate per hour
= 1
1
3
× £6.30 = £8.40
Let the number of overtime hours worked = x
Then,
(36)(6.30) + (x)(8.40) = 319.20
226.80 + 8.40x = 319.20