Higher Engineering Mathematics

TOTAL DIFFERENTIAL, RATES OF CHANGE AND SMALL CHANGES 351

G

Since the height is increasing at 3 mm/s,

i.e. 0.3 cm/s, then

dh

dt

=+ 0. 3

and since the radius is decreasing at 2 mm/s,

i.e. 0.2 cm/s, then

dr

dt

=− 0. 2

Hence

dV

dt

=

(

2

3

πrh

)

(− 0 .2)+

(

1

3

πr^2

)

(+ 0 .3)

=

− 0. 4

3

πrh+ 0. 1 πr^2

However, h= 3 .2 cm andr= 1 .5cm.

Hence

dV

dt

=

− 0. 4

3

π(1.5)(3.2)+(0.1)π(1.5)^2

=− 2. 011 + 0. 707 =− 1 .304 cm^3 /s

Thus the rate of change of volume is 1.30 cm^3 /s

decreasing.

Problem 6. The areaAof a triangle is given

byA=^12 acsinB, whereBis the angle between

sidesaandc.Ifais increasing at 0.4 units/s,c

is decreasing at 0.8 units/s andBis increasing at

0.2 units/s, find the rate of change of the area of

the triangle, correct to 3 significant figures, when

ais 3 units,cis 4 units andBisπ/6 radians.

Using equation (2), the rate of change of area,

dA

dt

=

∂A

∂a

da

dt

+

∂A

∂c

dc

dt

+

∂A

∂B

dB

dt

Since A=

1

2

acsinB,

∂A

∂a

=

1

2

csinB,

∂A

∂c

=

1

2

asinBand

∂A

∂B

=

1

2

accosB

da

dt

= 0 .4 units/s,

dc

dt

=− 0 .8 units/s

and

dB

dt

= 0 .2 units/s

Hence

dA

dt

=

(

1

2

csinB

)

(0.4)+

(

1

2

asinB

)

(− 0 .8)

+

(

1

2

accosB

)

(0.2)

Whena=3,c=4 andB=

π

6

then:

dA

dt

=

(

1

2

(4) sin

π

6

)

(0.4)+

(

1

2

(3) sin

π

6

)

(− 0 .8)

+

(

1

2

(3)(4) cos

π

6

)

(0.2)

= 0. 4 − 0. 6 + 1. 039 = 0 .839 units^2 /s, correct

to 3 significant figures.

Problem 7. Determine the rate of increase of

diagonalACof the rectangular solid, shown in

Fig. 35.1, correct to 2 significant figures, if the

sidesx,yandzincrease at 6 mm/s, 5 mm/s and

4 mm/s when these three sides are 5 cm, 4 cm

and 3 cm respectively.

y = 4 cm x = 5 cm

B

C

b

z = 3 cm

A

Figure 35.1

DiagonalAB=

√

(x^2 +y^2 )

DiagonalAC=

√

(BC^2 +AB^2 )

=

√

[z^2 +{

√

(x^2 +y^2 )}^2

=

√

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

LetAC=b, thenb=

√

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

Using equation (2), the rate of change of diagonalb

is given by:

db

dt

=

∂b

∂x

dx

dt

+

∂b

∂y

dy

dt

+

∂b

∂z

dz

dt

Sinceb=

√

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

∂b

∂x

=

1

2

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

− 1

(^2) (2x)=
x
√
(x^2 +y^2 +z^2 )