30 Basic Engineering Mathematics
Problem 26. The volume V cm
3 of a right
circular cone is given by V =
1
3
πr 2 h. Given that
radius r = 2.45 cm and height h = 18.7 cm, find the
volume, correct to 4 significant figures
V =
1
3
πr
2 h =
1
3
π(2.45)
2
(18.7)
=
1
3
× π × 2.45
2
× 18.7
= 117.544521 ...
Hence, volume, V = 117.5cm 3 , correct to 4 significant figures.
Problem 27. Force F newtons is given by the
formula F =
Gm 1 m 2
d 2 , where m 1 and m 2 are
masses, d their distance apart and G is a constant.
Find the value of the force given that
G = 6.67 × 10 −11 , m 1 = 7.36, m 2 = 15.5 and
d = 22.6. Express the answer in standard form,
correct to 3 significant figures
F =
Gm 1 m 2
d 2
=
(6.67 × 10 −11 )(7.36)(15.5)
(22.6) 2
=
(6.67)(7.36)(15.5)
(10 11 )(510.76)
=
1.490
10 11
Hence, force F = 1.49×10 −11 newtons, correct to
3 significant figures.
Problem 28. The time of swing, t seconds, of a
simple pendulum is given by t = 2π
l
g
Determine the time, correct to 3 decimal places,
given that l = 12.9 and g = 9.81
t = 2π
l
g
= (2π)
12.9
9.81
= 7.20510343 ...
Hence, time t = 7.205 seconds, correct to 3 decimal
places.
Problem 29. Resistance, R , varies with
temperature according to the formula
R = R 0 (1 + αt ). Evaluate R, correct to 3 significant
figures, given R 0 = 14.59, α = 0.0043 and t = 80
R = R 0 (1 + αt ) = 14.59[1 + (0.0043)(80)]
= 14.59(1 + 0.344)
= 14.59(1.344)
Hence, resistance, R = 19.6 , correct to 3 significant
figures.
Problem 30. The current, I amperes, in an a.c.
circuit is given by I =
V
(R 2 + X 2 )
. Evaluate the
current, correct to 2 decimal places, when
V = 250 V, R = 25.0 and X = 18.0 .
I =
V
(R 2 + X 2 )
=
250
25.0 2 + 18.0 2
= 8.11534341 ...
Hence, current, I = 8.12 A, correct to 2 decimal
places.
Now try the following Practice Exercise
Practice Exercise 20 Evaluation of
formulae (answers on page 341)
1. Find the total cost of 37 calculators costing £12.65 each and 19 drawing sets costing
£6.38 each.
2. Power =
force × distance
time
. Find the power
when a force of 3760 N raises an object a
distance of 4.73 m in 35 s.
3. The potential difference, V volts, available
at battery terminals is given by V = E − Ir.
Evaluate V when E = 5.62, I = 0.70 and
R = 4.30
4. Given force F =
1
2
m(v 2 − u 2 ), find F when
m = 18.3, v = 12.7 and u = 8.24
5. The current I amperes flowing in a number
of cells is given by I =
n E
R + nr
. Evaluate the
current when n = 36, E = 2.20, R = 2.80
and r = 0.50
6. The time, t seconds, of oscillation for a
simple pendulum is given by t = 2π
l
g
.
Determine the time when l = 54.32 and
g = 9.81
Problem 26. The volume V cm
3 of a right
circular cone is given by V =
1
3
πr 2 h. Given that
radius r = 2.45 cm and height h = 18.7 cm, find the
volume, correct to 4 significant figures
V =
1
3
πr
2 h =
1
3
π(2.45)
2
(18.7)
=
1
3
× π × 2.45
2
× 18.7
= 117.544521 ...
Hence, volume, V = 117.5cm 3 , correct to 4 significant figures.
Problem 27. Force F newtons is given by the
formula F =
Gm 1 m 2
d 2 , where m 1 and m 2 are
masses, d their distance apart and G is a constant.
Find the value of the force given that
G = 6.67 × 10 −11 , m 1 = 7.36, m 2 = 15.5 and
d = 22.6. Express the answer in standard form,
correct to 3 significant figures
F =
Gm 1 m 2
d 2
=
(6.67 × 10 −11 )(7.36)(15.5)
(22.6) 2
=
(6.67)(7.36)(15.5)
(10 11 )(510.76)
=
1.490
10 11
Hence, force F = 1.49×10 −11 newtons, correct to
3 significant figures.
Problem 28. The time of swing, t seconds, of a
simple pendulum is given by t = 2π
l
g
Determine the time, correct to 3 decimal places,
given that l = 12.9 and g = 9.81
t = 2π
l
g
= (2π)
12.9
9.81
= 7.20510343 ...
Hence, time t = 7.205 seconds, correct to 3 decimal
places.
Problem 29. Resistance, R , varies with
temperature according to the formula
R = R 0 (1 + αt ). Evaluate R, correct to 3 significant
figures, given R 0 = 14.59, α = 0.0043 and t = 80
R = R 0 (1 + αt ) = 14.59[1 + (0.0043)(80)]
= 14.59(1 + 0.344)
= 14.59(1.344)
Hence, resistance, R = 19.6 , correct to 3 significant
figures.
Problem 30. The current, I amperes, in an a.c.
circuit is given by I =
V
(R 2 + X 2 )
. Evaluate the
current, correct to 2 decimal places, when
V = 250 V, R = 25.0 and X = 18.0 .
I =
V
(R 2 + X 2 )
=
250
25.0 2 + 18.0 2
= 8.11534341 ...
Hence, current, I = 8.12 A, correct to 2 decimal
places.
Now try the following Practice Exercise
Practice Exercise 20 Evaluation of
formulae (answers on page 341)
1. Find the total cost of 37 calculators costing £12.65 each and 19 drawing sets costing
£6.38 each.
2. Power =
force × distance
time
. Find the power
when a force of 3760 N raises an object a
distance of 4.73 m in 35 s.
3. The potential difference, V volts, available
at battery terminals is given by V = E − Ir.
Evaluate V when E = 5.62, I = 0.70 and
R = 4.30
4. Given force F =
1
2
m(v 2 − u 2 ), find F when
m = 18.3, v = 12.7 and u = 8.24
5. The current I amperes flowing in a number
of cells is given by I =
n E
R + nr
. Evaluate the
current when n = 36, E = 2.20, R = 2.80
and r = 0.50
6. The time, t seconds, of oscillation for a
simple pendulum is given by t = 2π
l
g
.
Determine the time when l = 54.32 and
g = 9.81