78 Basic Engineering Mathematics
Using a calculator, gas constant, R = 4160 J/(kg K),
correct to 4 significant figures.
Problem 20. A rectangular box with square ends
has its length 15 cm greater than its breadth and the
total length of its edges is 2.04 m. Find the width of
the box and its volume
Let x cm = width = height of box. Then the length of
the box is (x + 15) cm, as shown in Figure 11.1.
( x 1
1 5 )
x
x
Figure 11.1
The length of the edges of the box is 2(4x) + 4(x + 15)
cm, which equals 2.04 m or 204 cm.
Hence,
204 = 2(4x) + 4(x + 15)
204 = 8x + 4x + 60
204 − 60 = 12x
i.e.
144 = 12x
and
x = 12 cm
Hence, the width of the box is 12 cm.
Volume of box = length × width × height
= (x + 15)(x)(x) = (12 + 15)(12)(12)
= (27)(12)(12)
= 3888 cm
3
Problem 21. The temperature coefficient of
resistance α may be calculated from the formula
R t = R 0 (1 + αt ). Find α, given R t = 0.928,
R 0 = 0.80 and t = 40
Since R t = R 0 (1 + αt ), then
0.928 = 0.80[1 + α(40)]
0.928 = 0.80 + (0.8)(α)(40)
0.928 − 0.80 = 32α
0.128 = 32α
Hence,
α =
0.128
32
= 0.004
Problem 22. The distance s metres travelled
in time t seconds is given by the formula
s = ut +
1
2
at
2 , where u is the initial velocity in m/s
and a is the acceleration in m/s 2 . Find the
acceleration of the body if it travels 168 m in 6 s,
with an initial velocity of 10 m/s
s = ut +
1
2
at
2
, and s = 168, u = 10 and t = 6
Hence,
168 = (10)(6) +
1
2
a(6)
2
168 = 60 + 18a
168 − 60 = 18a
108 = 18a
a =
108
18
= 6
Hence, the acceleration of the body is 6 m/s 2 .
Problem 23. When three resistors in an electrical
circuit are connected in parallel the total resistance
R T is given by
1
R T
=
1
R 1
+
1
R 2
+
1
R 3
. Find the
total resistance when R 1 = 5 , R 2 = 10 and
R 3 = 30
1
R T
=
1
5
+
1
10
+
1
30
=
6 + 3 + 1
30
=
10
30
=
1
3
Taking the reciprocal of both sides gives R T = 3
Alternatively, if
1
R T
=
1
5
+
1
10
+
1
30
, the LCM of the
denominators is 30R T.
Hence, 30R T
1
R T
= 30R T
1
5
+ 30R T
1
10
+ 30R T
1
30
.
Cancelling gives 30 = 6R T + 3R T + R T
i.e.
30 = 10R T
and
R T =
30
10
= 3 , as above.
Using a calculator, gas constant, R = 4160 J/(kg K),
correct to 4 significant figures.
Problem 20. A rectangular box with square ends
has its length 15 cm greater than its breadth and the
total length of its edges is 2.04 m. Find the width of
the box and its volume
Let x cm = width = height of box. Then the length of
the box is (x + 15) cm, as shown in Figure 11.1.
( x 1
1 5 )
x
x
Figure 11.1
The length of the edges of the box is 2(4x) + 4(x + 15)
cm, which equals 2.04 m or 204 cm.
Hence,
204 = 2(4x) + 4(x + 15)
204 = 8x + 4x + 60
204 − 60 = 12x
i.e.
144 = 12x
and
x = 12 cm
Hence, the width of the box is 12 cm.
Volume of box = length × width × height
= (x + 15)(x)(x) = (12 + 15)(12)(12)
= (27)(12)(12)
= 3888 cm
3
Problem 21. The temperature coefficient of
resistance α may be calculated from the formula
R t = R 0 (1 + αt ). Find α, given R t = 0.928,
R 0 = 0.80 and t = 40
Since R t = R 0 (1 + αt ), then
0.928 = 0.80[1 + α(40)]
0.928 = 0.80 + (0.8)(α)(40)
0.928 − 0.80 = 32α
0.128 = 32α
Hence,
α =
0.128
32
= 0.004
Problem 22. The distance s metres travelled
in time t seconds is given by the formula
s = ut +
1
2
at
2 , where u is the initial velocity in m/s
and a is the acceleration in m/s 2 . Find the
acceleration of the body if it travels 168 m in 6 s,
with an initial velocity of 10 m/s
s = ut +
1
2
at
2
, and s = 168, u = 10 and t = 6
Hence,
168 = (10)(6) +
1
2
a(6)
2
168 = 60 + 18a
168 − 60 = 18a
108 = 18a
a =
108
18
= 6
Hence, the acceleration of the body is 6 m/s 2 .
Problem 23. When three resistors in an electrical
circuit are connected in parallel the total resistance
R T is given by
1
R T
=
1
R 1
+
1
R 2
+
1
R 3
. Find the
total resistance when R 1 = 5 , R 2 = 10 and
R 3 = 30
1
R T
=
1
5
+
1
10
+
1
30
=
6 + 3 + 1
30
=
10
30
=
1
3
Taking the reciprocal of both sides gives R T = 3
Alternatively, if
1
R T
=
1
5
+
1
10
+
1
30
, the LCM of the
denominators is 30R T.
Hence, 30R T
1
R T
= 30R T
1
5
+ 30R T
1
10
+ 30R T
1
30
.
Cancelling gives 30 = 6R T + 3R T + R T
i.e.
30 = 10R T
and
R T =
30
10
= 3 , as above.