Understanding Engineering Mathematics

(i) An oil spill from a ruptured tanker in calm seas spreads out in a circular

pattern with the radius increasing at a constant rate of 1 ms−^1 .How

fast is the area of the spill increasing when the radius is 300 m?

(ii) The radius of a spherical balloon is increasing at 0.001 ms−^1 .Atwhat

rate is the (i) surface area (ii) volume increasing when the radius is

0.25 m?

3.The motion of a particle performing damped vibrations is given by

x=e−tsin 2t

wherexis the displacement in metres from its mean position at timetsecs. Determine

the times at whichxis a maximum and find the maximum distance for the least positive

value oft. Determine the acceleration at this point.

4.The work done in an air compressor is given by

W=K





(

p 1

p

)n−^1

n

+

(

p

p 2

)n−^1

n

− 2





wherep 1 ,p 2 ,n,Kare positive constants. Show that the work done is a minimum

whenp=

√

p 1 p 2.

5.One method of obtaining an estimatex ̄of a quantityxfrom a set ofnmeasurements
ofx(all of which may be subject to experimental error),x 1 ,x 2 ,...,xnis to choosex ̄
to minimise the sum:

s=(x 1 − ̄x)^2 +(x 2 − ̄x)^2 +···+(xn− ̄x)^2

This is called themethod of least squares. It minimises the total squared deviation

fromx ̄.

Determinex ̄according to this principle and comment on the result.

6.The resultant mass of a system of particles, massesmi, situated at the pointsPi(xi,yi,zi)
acts at a fixed pointG(x, ̄ y, ̄ z) ̄ where

x ̄=

∑

mixi

M

y ̄=

∑

miyi

M

z ̄=

∑

mizi

M

whereM=

∑

mi

Gis called thecentre of mass. The numerators in the above expressions are called

the first moments of the system with respect to theyz-,zx-,xy-planes respectively. In

the case of a continuous body the above particle system is replaced by the elements

of the body and integrals used as the limits of the summations. For example:

x ̄=

∫

xdm

∫

dm

The centre of massGof a plane lamina lies in the plane of the lamina. If the lamina is

uniform,Gis called thecentroidof the area of the lamina and if this area is symmetrical

about any straight line, the centroid lies on this line. The coordinates of the centre of