In this problem we will consider nutrient absorption and consumption in cells of

Question

In this problem we will consider nutrient absorption and consumption in cells of a different shape. Namely, we assume that each cell may be represented using a solid made up of a hemispherical of radius r joined to the flat side of a cone with radius r and height 3r as shown below. Assume as in Section 1.3 that the rate of absorption of nutrients is proportional to the surface area of the cell and the rate of consumption is proportional to the volume of the cell. (a) Find formulas for the volume and surface area as functions of r (b) Find formulas for the rate of absorption and rate of consumption as a function of r (introduce necessary constants) (c) What is the maximum r value for which these cells can survive?

Explanation / Answer

(a) volume of cell = volume of cone + volume of hemisphere

Volume of cone = (1/3)**(radius)2*(height)

= (1/3)**r2*(3r)

Volume of cone = .r3

volume of hemisphere = (2/3)* *r3

Therefore

Volume of cell = (5/3)**r3.

Similarly

surface area of cell = surface area of ( cone + hemisphere)

Surface area of cone = *(radius)*(slant height)

= *r*(10)*r

=*(10)*r2

Surface area of hemisphere = 2*r2

Therefore

surface area of cell = (2+10)*.r2

(b) given rate of absorption proportional to surface area

i.e)

rate of absorption = k*(surface area)

rate of absorption = k*(2+10)*r2

similarly

rate of consumption = K*(5/3)*r3

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