Higher Engineering Mathematics

SOME APPLICATIONS OF DIFFERENTIATION 309

G

Figure 28.11

Substituting into equation (1) gives:

V=π

(

144 −

h^2

4

)

h= 144 πh−

πh^3

4

dV

dh

= 144 π−

3 πh^2

4

=0,

for a maximum or minimum value.
Hence

144 π=

3 πh^2

4

from which, h=

√

(144)(4)

3

= 13 .86 cm

d^2 V

dh^2

=

− 6 πh

4

Whenh= 13 .86,

d^2 V

dh^2

is negative, giving a maxi-

mum value.

From equation (2),

r^2 = 144 −

h^2

4

= 144 −

13. 862

4

from which, radiusr= 9 .80 cm

Diameter of cylinder= 2 r=2(9.80)= 19 .60 cm.

Hence the cylinder having the maximum volume
that can be cut from a sphere of radius 12 cm
is one in which the diameter is 19.60 cm and the
height is 13.86 cm.

Now try the following exercise.

Exercise 125 Further problems on practical

maximum and minimum problems

1. The speed,v, of a car (in m/s) is related to
   timetsby the equationv= 3 + 12 t− 3 t^2.
   Determine the maximum speed of the car
   in km/h. [54 km/h]


2. Determine the maximum area of a rectangu-
   lar piece of land that can be enclosed by
   1200 m of fencing. [90000 m^2 ]


3. A shell is fired vertically upwards and
   its vertical height, xmetres, is given by
   x= 24 t− 3 t^2 , wheretis the time in seconds.
   Determine the maximum height reached.
   [48 m]


4. A lidless box with square ends is to be made
   from a thin sheet of metal. Determine the
   least area of the metal for which the volume
   of the box is 3.5 m^3. [11.42 m^2 ]


5. A closed cylindrical container has a surface
   area of 400 cm^2. Determine the dimensions
   for maximum volume.[
   radius= 4 .607 cm;
   height= 9 .212 cm

]

6. Calculate the height of a cylinder of max-
   imum volume which can be cut from a cone
   of height 20 cm and base radius 80 cm.
   [6.67 cm]


7. The power developed in a resistorRby a
   battery of emfEand internal resistanceris

given byP=

E^2 R

(R+r)^2

. DifferentiatePwith

respect toRand show that the power is a

maximum whenR=r.

8. Find the height and radius of a closed cylin-
   der of volume 125 cm^3 which has the least
   surface area. [
   height= 5 .42 cm;
   radius= 2 .71 cm

]

9. Resistance to motion,F, of a moving vehi-
   cle, is given byF=^5 x+ 100 x. Determine the
   minimum value of resistance. [44.72]