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Problem 53 Easy Difficulty

Let $ A(t) $ be the area of a tissue culture at time $ t $ and let $ M $ be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery is proportional to $ \sqrt {A(t)}. $ So a reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to $ \sqrt {A(t)} $ and $ M - A(t). $
(a) Formulate a differential equation and use it to show that the tissue grows fastest when $ A(t) = \frac {1}{3} M. $
(b) Solve the differential equation to find an expression for $ A(t). $ Use a computer algebra system to perform the integration.

Answer

a) $$=\frac{1}{2} k A^{-1 / 2}[k \sqrt{A}(M-A)][M-3 A]=\frac{1}{2} k^{2}(M-A)(M-3 A)$$
b) $$C=\frac{\sqrt{M}+\sqrt{A_{0}}}{\sqrt{M}-\sqrt{A_{0}}}$$

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