00:01
So given in the question, the volume v of a growing spherical cell is given by the expression v is equal to 4 by 3 pi r cube where r is the radius of the cell measured in micrometers.
00:27
We know one micrometer is 10 to the core minus 6 meters.
00:31
It has been asked to find part a average rate of change of we with respect to r when r changes from two micrometers to the following first from two micrometers to five micrometers from two micrometers second from two micrometers to three micrometers and thirds from 2 micrometers to 2 .1 micrometers.
01:14
And the b part of the question is to find the instantaneous rate of change of volume v with respect to r when r is 2 micrmometers.
01:28
So now solving the question, we know the average rate of change of volume v with respect to r, when the radius changes from, r equal to r1 to r2 is given as follows.
01:59
The delta v by delta r that will be equal to v at r2 minus v at r1 divided by r2 minus r1 which is 4 by r2 2 by r2 cube minus 4 by r1 cube divided by r2 minus r2.
02:25
And the instantaneous rate of change of v with respect to r1, at any value of r is obtained by differentiating v with respect to r and then by the putting the value of r in that expression.
02:48
So dv by d r is d by d r times 4 by 3 pi a cube which is 4 pi r square.
03:00
So solving the first a part a, the first section of part a when r1 is 2 micrometers and r2 is 5 micrometers, delta v by delta r would be 4 by 3 pi 5 micrometer cube minus 4 by 3 pi 2 micrometer cubed divided by 5 micrometer minus 2 micrometers.
03:29
So that comes out to be 4 by 3 pi and is 5 cube is 125 minus 8 divided by 5 minus 2 is 3 micrometer cube by micrometers so which is 117 by 3 so dividing it with 3 we get 39 and then again with 3 we get 13 so finally we get 52 pi micrometer cube by micro putting the value of pi as 3 .14 we get the expression to be 163 .28 which upon restricting to first place of decimal we get it to be 63.
04:16
3 .3 micrometer cube by micrometer...