356 Higher Engineering Mathematics
Using equation (3), the approximate change in t ,
δt ≈
∂t
∂l
δl +
∂t
∂g
δg
Since t = 2π
l
g
,
∂t
∂l
=
π
√
lg
and
∂t
∂g
= −π
l
g 3 (from Problem 6, Chapter 34)
δl =
0.2
100
l = 0.002 l and δg = −0.001g
hence δt ≈
π
√
lg
(0.002l) + −π
l
g 3 (−0.001 g)
≈ 0.002π
l
g
+ 0.001π
l
g
≈ (0.001)
2π
l
g
+ 0.0005
2π
l
g
≈ 0.0015t ≈
0.15
100
t
Hence the approximate error in t is a 0.15% increase.
Now try the following exercise
Exercise 142 Further problems on small
changes
1. The power P consumed in a resistor is given by
P = V 2 /R watts. Determine the approximate
change in power when V increases by 5% and
R decreases by 0.5% if the original values of V
and R are 50 volts and 12.5 ohms respectively.
[+21 watts]
2. An equation for heat generated H is H = i 2 Rt .
Determine the error in the calculated value of
H if the error in measuring current i is +2%,
the error in measuring resistance R is −3%
and the error in measuring time t is +1%.
[+2%]
3. f r =
1
2π
√
LC
represents the resonant
frequency of a series connected circuit
containing inductance L and capacitance C.
Determine the approximate percentage
change in f r when L is decreased by 3% and
C is increased by 5%.
[−1%]
4. The second moment of area of a rectangle
about its centroid parallel to side b is given by
I = bd 3 /12. If b and d are measured as 15 cm
and 6 cm respectively and the measurement
errors are +12 mm in b and −1.5 mm in d,
find the error in the calculated value of I .
[+1.35 cm 4 ]
5. The side b of a triangle is calculated using
b
2
= a
2
+ c
2
− 2ac cos B. If a, c and B are
measured as 3 cm, 4 cm and π/4 radians respectively and the measurement errors
which occur are +0.8 cm, −0.5 cm and +π/90
radians respectively, determine the error in the
calculated value of b.
[−0.179 cm]
6. Q factor in a resonant electrical circuit is given
by: Q =
1
R
L
C
. Find the percentage change in
Q when L increases by 4%, R decreases by 3%
and C decreases by 2%.
[+6%]
7. The rate of flow of gas in a pipe is given by:
v =
C
√
d
6
√
T 5
, where C is a constant, d is the diameter of the pipe and T is the thermodynamic
temperature of the gas. When determining the
rate of flow experimentally, d is measured and
subsequently found to be in error by +1.4%,
and T has an error of −1.8%. Determine the
percentage error in the rate of flow based on
the measured values of d and T .
[+2.2%]
Using equation (3), the approximate change in t ,
δt ≈
∂t
∂l
δl +
∂t
∂g
δg
Since t = 2π
l
g
,
∂t
∂l
=
π
√
lg
and
∂t
∂g
= −π
l
g 3 (from Problem 6, Chapter 34)
δl =
0.2
100
l = 0.002 l and δg = −0.001g
hence δt ≈
π
√
lg
(0.002l) + −π
l
g 3 (−0.001 g)
≈ 0.002π
l
g
+ 0.001π
l
g
≈ (0.001)
2π
l
g
+ 0.0005
2π
l
g
≈ 0.0015t ≈
0.15
100
t
Hence the approximate error in t is a 0.15% increase.
Now try the following exercise
Exercise 142 Further problems on small
changes
1. The power P consumed in a resistor is given by
P = V 2 /R watts. Determine the approximate
change in power when V increases by 5% and
R decreases by 0.5% if the original values of V
and R are 50 volts and 12.5 ohms respectively.
[+21 watts]
2. An equation for heat generated H is H = i 2 Rt .
Determine the error in the calculated value of
H if the error in measuring current i is +2%,
the error in measuring resistance R is −3%
and the error in measuring time t is +1%.
[+2%]
3. f r =
1
2π
√
LC
represents the resonant
frequency of a series connected circuit
containing inductance L and capacitance C.
Determine the approximate percentage
change in f r when L is decreased by 3% and
C is increased by 5%.
[−1%]
4. The second moment of area of a rectangle
about its centroid parallel to side b is given by
I = bd 3 /12. If b and d are measured as 15 cm
and 6 cm respectively and the measurement
errors are +12 mm in b and −1.5 mm in d,
find the error in the calculated value of I .
[+1.35 cm 4 ]
5. The side b of a triangle is calculated using
b
2
= a
2
+ c
2
− 2ac cos B. If a, c and B are
measured as 3 cm, 4 cm and π/4 radians respectively and the measurement errors
which occur are +0.8 cm, −0.5 cm and +π/90
radians respectively, determine the error in the
calculated value of b.
[−0.179 cm]
6. Q factor in a resonant electrical circuit is given
by: Q =
1
R
L
C
. Find the percentage change in
Q when L increases by 4%, R decreases by 3%
and C decreases by 2%.
[+6%]
7. The rate of flow of gas in a pipe is given by:
v =
C
√
d
6
√
T 5
, where C is a constant, d is the diameter of the pipe and T is the thermodynamic
temperature of the gas. When determining the
rate of flow experimentally, d is measured and
subsequently found to be in error by +1.4%,
and T has an error of −1.8%. Determine the
percentage error in the rate of flow based on
the measured values of d and T .
[+2.2%]