80 Basic Engineering Mathematics
8.40x = 319.20 − 226.80 = 92.40
x =
92.40
8.40
= 11
Thus, 11 hours overtime would have to be worked to
earn £319.20 per week. Hence, the total number of
hours worked is 36 + 11, i.e. 47 hours.
Problem 27. A formula relating initial and final
states of pressures, P 1 and P 2 , volumes, V 1 and V 2 ,
and absolute temperatures, T 1 and T 2 , of an ideal
gas is
P 1 V 1
T 1
=
P 2 V 2
T 2
. Find the value of P 2 given
P 1 = 100 × 10 3 , V 1 = 1.0, V 2 = 0.266, T 1 = 423
and T 2 = 293
Since
P 1 V 1
T 1
=
P 2 V 2
T 2
then
(100 × 10 3 )(1.0)
423
=
P 2 (0.266)
293
Cross-multiplying gives
(100 × 10
3
)(1.0)(293) = P 2 (0.266)(423)
P 2 =
(100 × 10 3 )(1.0)(293)
(0.266)(423)
Hence,
P 2 = 260 × 10
3 or 2.6 × 10
5
.
Problem 28. The stress, f , in a material of a
thick cylinder can be obtained from
D
d
=
f + p
f − p
. Calculate the stress, given that
D = 21.5, d = 10.75 and p = 1800
Since
D
d
=
f + p
f − p
then
21.5
10.75
=
f + 1800
f − 1800
i.e.
2 =
f + 1800
f − 1800
Squaring both sides gives
4 =
f + 1800
f − 1800
Cross-multiplying gives
4( f − 1800) = f + 1800
4 f − 7200 = f + 1800
4 f − f = 1800 + 7200
3 f = 9000
f =
9000
3
= 3000
Hence,
stress,f = 3000
Problem 29. 12 workmen employed on a
building site earn between them a total of £4035 per
week. Labourers are paid £275 per week and
craftsmen are paid £380 per week. How many
craftsmen and how many labourers are employed?
Let the number of craftsmen be c. The number of
labourers is therefore (12 − c).
The wage bill equation is
380c + 275(12 − c) = 4035
380c + 3300 − 275c = 4035
380c − 275c = 4035 − 3300
105c = 735
c =
735
105
= 7
Hence, there are 7 craftsmen and (12 − 7), i.e. 5
labourers on the site.
Now try the following Practice Exercise
Practice Exercise 45 Practical problems
involving simple equations (answers on
page 344)
1. A rectangle has a length of 20 cm and a width
b cm. When its width is reduced by 4 cm its
area becomes 160 cm 2 . Find the original width
and area of the rectangle.
2. Given R 2 = R 1 (1 + αt ), find α given
R 1 = 5.0, R 2 = 6.03 and t = 51.5
3. If v 2 = u 2 + 2as, find u given v = 24,
a = −40 and s = 4.05
4. The relationship between the temperature on
a Fahrenheit scale and that on a Celsius scale
is given by F =
9
5
C + 32. Express 113 ◦ F in
degrees Celsius.
5. If t = 2π
w
Sg
, find the value of S given
w = 1.219, g = 9.81 and t = 0.3132
6. Two joiners and five mates earn £1824
between them for a particular job. If a joiner
earns £72 more than a mate, calculate the
earnings for a joiner and for a mate.
8.40x = 319.20 − 226.80 = 92.40
x =
92.40
8.40
= 11
Thus, 11 hours overtime would have to be worked to
earn £319.20 per week. Hence, the total number of
hours worked is 36 + 11, i.e. 47 hours.
Problem 27. A formula relating initial and final
states of pressures, P 1 and P 2 , volumes, V 1 and V 2 ,
and absolute temperatures, T 1 and T 2 , of an ideal
gas is
P 1 V 1
T 1
=
P 2 V 2
T 2
. Find the value of P 2 given
P 1 = 100 × 10 3 , V 1 = 1.0, V 2 = 0.266, T 1 = 423
and T 2 = 293
Since
P 1 V 1
T 1
=
P 2 V 2
T 2
then
(100 × 10 3 )(1.0)
423
=
P 2 (0.266)
293
Cross-multiplying gives
(100 × 10
3
)(1.0)(293) = P 2 (0.266)(423)
P 2 =
(100 × 10 3 )(1.0)(293)
(0.266)(423)
Hence,
P 2 = 260 × 10
3 or 2.6 × 10
5
.
Problem 28. The stress, f , in a material of a
thick cylinder can be obtained from
D
d
=
f + p
f − p
. Calculate the stress, given that
D = 21.5, d = 10.75 and p = 1800
Since
D
d
=
f + p
f − p
then
21.5
10.75
=
f + 1800
f − 1800
i.e.
2 =
f + 1800
f − 1800
Squaring both sides gives
4 =
f + 1800
f − 1800
Cross-multiplying gives
4( f − 1800) = f + 1800
4 f − 7200 = f + 1800
4 f − f = 1800 + 7200
3 f = 9000
f =
9000
3
= 3000
Hence,
stress,f = 3000
Problem 29. 12 workmen employed on a
building site earn between them a total of £4035 per
week. Labourers are paid £275 per week and
craftsmen are paid £380 per week. How many
craftsmen and how many labourers are employed?
Let the number of craftsmen be c. The number of
labourers is therefore (12 − c).
The wage bill equation is
380c + 275(12 − c) = 4035
380c + 3300 − 275c = 4035
380c − 275c = 4035 − 3300
105c = 735
c =
735
105
= 7
Hence, there are 7 craftsmen and (12 − 7), i.e. 5
labourers on the site.
Now try the following Practice Exercise
Practice Exercise 45 Practical problems
involving simple equations (answers on
page 344)
1. A rectangle has a length of 20 cm and a width
b cm. When its width is reduced by 4 cm its
area becomes 160 cm 2 . Find the original width
and area of the rectangle.
2. Given R 2 = R 1 (1 + αt ), find α given
R 1 = 5.0, R 2 = 6.03 and t = 51.5
3. If v 2 = u 2 + 2as, find u given v = 24,
a = −40 and s = 4.05
4. The relationship between the temperature on
a Fahrenheit scale and that on a Celsius scale
is given by F =
9
5
C + 32. Express 113 ◦ F in
degrees Celsius.
5. If t = 2π
w
Sg
, find the value of S given
w = 1.219, g = 9.81 and t = 0.3132
6. Two joiners and five mates earn £1824
between them for a particular job. If a joiner
earns £72 more than a mate, calculate the
earnings for a joiner and for a mate.