66 Higher Engineering Mathematics
Now try the following exercise
Exercise 31 Further practical problems
involving the binomial theorem
1. Pressure p and volume v are related by
pv 3 = c, where c is a constant. Determine the
approximate percentage change in c when
p is increased by 3% and v decreased by
1.2%.
[0.6% decrease]
2. Kinetic energy is given by
1
2 mv 2 . Determine
the approximate change in the kinetic energy
when mass m is increased by 2.5% and the
velocity v is reduced by 3%.
[3.5% decrease]
3. An error of +1.5% was made when measuring the radius of a sphere. Ignoring the
products of small quantities determine the
approximate error in calculating (a) the volume, and (b) the surface area.
(a) 4.5% increase
(b) 3.0% increase
4. The power developed by an engine is given
by I = k PLAN, where k is a constant. Determine the approximate percentage change in
the power when P and A are each increased
by 2.5% and L and N are each decreased
by 1.4%.
[2.2% increase]
5. The radius of a cone is increased by 2.7%
and its height reduced by 0.9%. Determine
the approximate percentage change in its
volume, neglecting the products of small
terms.
[4.5% increase]
6. The electric field strength H due to a magnet
of length 2l and moment M at a point on its
axis distance x from the centre is given by
H =
M
2l
1
(x − l) 2 −
1
(x + l) 2
Show that if l is very small compared with x,
then H ≈
2M
x 3 .
7. The shear stress τ in a shaft of diameter
D under a torque T is given by: τ =
kT
π D 3 .
Determine the approximate percentage error
in calculating τ if T is measured 3% too small
and D 1.5% too large.
[7.5% decrease]
8. The energy W stored in a flywheel is given
by: W = kr 5 N 2 , where k is a constant, r
is the radius and N the number of revolutions. Determine the approximate percentage
change in W when r is increased by 1.3% and
N is decreased by 2%.
[2.5% increase]
9. In a series electrical circuit containing inductance L and capacitance C the resonant frequency is given by: f r =
1
2π
√
LC
. If the
values of L and C used in the calculation are
2.6% too large and 0.8% too small respectively, determine the approximate percentage
error in the frequency.
[0.9% too small]
10. The viscosity η of a liquid is given by:
η =
kr 4
νl
, where k is a constant. If there is
an error in r of +2%, in ν of +4% and l of
−3%, what is the resultant error in η?
[+7%]
11. A magnetic pole, distance x from the plane
of a coil of radius r, and on the axis of the
coil, is subject to a force F when a current flows in the coil. The force is given by:
F =
kx
(r 2 + x 2 ) 5
, where k is a constant. Use
the binomial theorem to show that when x is
small compared to r, then
F ≈
kx
r 5 −
5kx 3
2r 7 .
12. The flow of water through a pipe is given by:
G =
(3d) 5 H
L
. If d decreases by 2% and H
by 1%, use the binomial theorem to estimate
the decrease in G.
[5.5%]
Now try the following exercise
Exercise 31 Further practical problems
involving the binomial theorem
1. Pressure p and volume v are related by
pv 3 = c, where c is a constant. Determine the
approximate percentage change in c when
p is increased by 3% and v decreased by
1.2%.
[0.6% decrease]
2. Kinetic energy is given by
1
2 mv 2 . Determine
the approximate change in the kinetic energy
when mass m is increased by 2.5% and the
velocity v is reduced by 3%.
[3.5% decrease]
3. An error of +1.5% was made when measuring the radius of a sphere. Ignoring the
products of small quantities determine the
approximate error in calculating (a) the volume, and (b) the surface area.
(a) 4.5% increase
(b) 3.0% increase
4. The power developed by an engine is given
by I = k PLAN, where k is a constant. Determine the approximate percentage change in
the power when P and A are each increased
by 2.5% and L and N are each decreased
by 1.4%.
[2.2% increase]
5. The radius of a cone is increased by 2.7%
and its height reduced by 0.9%. Determine
the approximate percentage change in its
volume, neglecting the products of small
terms.
[4.5% increase]
6. The electric field strength H due to a magnet
of length 2l and moment M at a point on its
axis distance x from the centre is given by
H =
M
2l
1
(x − l) 2 −
1
(x + l) 2
Show that if l is very small compared with x,
then H ≈
2M
x 3 .
7. The shear stress τ in a shaft of diameter
D under a torque T is given by: τ =
kT
π D 3 .
Determine the approximate percentage error
in calculating τ if T is measured 3% too small
and D 1.5% too large.
[7.5% decrease]
8. The energy W stored in a flywheel is given
by: W = kr 5 N 2 , where k is a constant, r
is the radius and N the number of revolutions. Determine the approximate percentage
change in W when r is increased by 1.3% and
N is decreased by 2%.
[2.5% increase]
9. In a series electrical circuit containing inductance L and capacitance C the resonant frequency is given by: f r =
1
2π
√
LC
. If the
values of L and C used in the calculation are
2.6% too large and 0.8% too small respectively, determine the approximate percentage
error in the frequency.
[0.9% too small]
10. The viscosity η of a liquid is given by:
η =
kr 4
νl
, where k is a constant. If there is
an error in r of +2%, in ν of +4% and l of
−3%, what is the resultant error in η?
[+7%]
11. A magnetic pole, distance x from the plane
of a coil of radius r, and on the axis of the
coil, is subject to a force F when a current flows in the coil. The force is given by:
F =
kx
(r 2 + x 2 ) 5
, where k is a constant. Use
the binomial theorem to show that when x is
small compared to r, then
F ≈
kx
r 5 −
5kx 3
2r 7 .
12. The flow of water through a pipe is given by:
G =
(3d) 5 H
L
. If d decreases by 2% and H
by 1%, use the binomial theorem to estimate
the decrease in G.
[5.5%]