In this problem we will consider capsule shaped cells and use an alternate mathe

Question

In this problem we will consider capsule shaped cells and use an alternate mathematical model. Namely, we will assume that each cell may be represented using a solid made up of a cylinder of radius r and height 3r together with a hemispherical cap on each end as shown below. Assume as in Section 1.2 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. Find formulas for the volume and surface area of this shape as a function of r Find formulas for the rate of absorption and rate of consumption as a function of r (introduce necessary constants) What is the maximum r value for which these cells can survive?

Explanation / Answer

a> Surface area of the capsule S(r) = surface area of the cylinder + 2*surface area of the hemisphere

                                                                  = 2pi*radius*height + 2*2pi*radius^2

                                                                 = 2pi*r*3r + 4pir^2

                                                                 = 6*pi*r^2 + 4*pi*r^2 = 10*pi*r^2

Volumevof the capsule = V(r) = volume of the cylinder + 2*volume of the Hemisphere

                                                     = pi*r^2*(3r) + 2*2/3*pi*r^3

                                                     = 3pi*r^3 + 4/3*pi*r^3

                                                     = 13/3*pi*r^3

b> Let dA/dt represent the rate of absorption

=> dA/dt = K1*10*pi*r^2 , -----------> K1 is the proportionality constant

Let dC/dt represent the rate of consumption

=> dC/dt = K2*13/3*pi*r^3 , K2 is the proportionality constant

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