Higher Engineering Mathematics

352 DIFFERENTIAL CALCULUS

Similarly,

∂b

∂y

=

y

√

(x^2 +y^2 +z^2 )

and

∂b

∂z

=

z

√

(x^2 +y^2 +z^2 )

dx

dt

=6 mm/s= 0 .6 cm/s,

dy

dt

=5 mm/s= 0 .5 cm/s,

and

dz

dt

=4 mm/s= 0 .4 cm/s

Hence

db

dt

=

[

x

√

(x^2 +y^2 +z^2 )

]

(0.6)

+

[

y

√

(x^2 +y^2 +z^2 )

]

(0.5)

+

[

z

√

(x^2 +y^2 +z^2 )

]

(0.4)

Whenx=5 cm,y=4 cm andz=3 cm, then:

db

dt

=

[

5

√

(5^2 + 42 + 32 )

]

(0.6)

+

[

4

√

(5^2 + 42 + 32 )

]

(0.5)

+

[

3

√

(5^2 + 42 + 32 )

]

(0.4)

= 0. 4243 + 0. 2828 + 0. 1697 = 0 .8768 cm/s

Hence the rate of increase of diagonalAC is

0.88 cm/s or 8.8 mm/s, correct to 2 significant

figures.

Now try the following exercise.

Exercise 142 Further problems on rates of

change

1. The radius of a right cylinder is increas-
   ing at a rate of 8 mm/s and the height is
   decreasing at a rate of 15 mm/s. Find the
   rate at which the volume is changing in

cm^3 /s when the radius is 40 mm and the

height is 150 mm. [+ 226 .2cm^3 /s]

2. Ifz=f(x,y) andz= 3 x^2 y^5 , find the rate of
   change ofzwhenxis 3 units andyis 2 units
   whenxis decreasing at 5 units/s andyis
   increasing at 2.5 units/s. [2520 units/s]


3. Find the rate of change of k, correct to
   4 significant figures, given the following
   data:k=f(a,b,c); k= 2 blna+c^2 ea; ais
   increasing at 2 cm/s; b is decreasing at
   3 cm/s;cis decreasing at 1 cm/s;a= 1 .5 cm,
   b=6 cm andc=8 cm. [515.5 cm/s]


4. A rectangular box has sides of lengthxcm,
   ycm andzcm. Sidesxandzare expanding
   at rates of 3 mm/s and 5 mm/s respectively
   and sideyis contracting at a rate of 2 mm/s.
   Determine the rate of change of volume when
   xis 3 cm,yis 1.5 cm andzis 6 cm.
   [1.35 cm^3 /s]


5. Find the rate of change of the total surface
   area of a right circular cone at the instant
   when the base radius is 5 cm and the height
   is 12 cm if the radius is increasing at 5 mm/s
   and the height is decreasing at 15 mm/s.
   [17.4 cm^2 /s]

35.3 Small changes

It is often useful to find an approximate value for

the change (or error) of a quantity caused by small

changes (or errors) in the variables associated with

the quantity. Ifz=f(u,v,w,...) andδu,δv,δw,...

denotesmall changes inu,v,w,...respectively,

then the corresponding approximate changeδzin

zis obtained from equation (1) by replacing the

differentials by the small changes.

Thus δz≈

∂z

∂u

δu+

∂z

∂v

δv+

∂z

∂w

δw+··· (3)

Problem 8. Pressurepand volumeVofagas

are connected by the equationpV^1.^4 =k. Deter-

mine the approximate percentage error inkwhen

the pressure is increased by 4% and the volume

is decreased by 1.5%.