Higher Engineering Mathematics, Sixth Edition

Total differential, rates of change and small changes 353

Hence the rate of change of z,

dz

dt

=75.14units/s,

correct to 4 significant figures.

Problem 5. The height of a right circular cone is

increasing at 3mm/s and its radius is decreasing at

2mm/s. Determine, correct to 3 significant figures,

the rate at which the volume is changing (in cm^3 /s)

when the height is 3.2cm and the radius is 1.5cm.

Volume of a right circular cone,V=

1

3

πr^2 h

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

dV

dt

=

∂V

∂r

dr

dt

+

∂V

∂h

dh

dt

∂V

∂r

=

2

3

πrhand

∂V

∂h

=

1

3

πr^2

Since the height is increasing at 3mm/s,

i.e. 0.3cm/s, then
dh
dt

=+ 0. 3

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

i.e. 0.2cm/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 .2cmandr= 1 .5cm.

Hence

dV

dt

=

− 0. 4

3

π( 1. 5 )( 3. 2 )+( 0. 1 )π( 1. 5 )^2

=− 2. 011 + 0. 707 =− 1 .304cm^3 /s

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

Problem 6. The areaAof a triangle is given by

A=^12 acsinB,whereBis the angle between sidesa

andc.Ifais increasing at 0.4 units/s,cis

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, whenais 3

units,cis 4 units andBisπ/6radians.

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=4andB=

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 .839units^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 sides

x,yandzincrease at 6mm/s, 5mm/s and 4mm/s

when these three sides are 5cm, 4cm and 3cm

respectively.

C

b

B z 5 3cm

x^5

y 5 5cm

4cm

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 )