Let A (t) be the area of a tissue culture at time and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to √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 √A (t) and M – A (t).
(a) Formulate a differential equation and use it to show that the tissue grows fastest when A (t) = M/3.
(b) Solve the differential equation to find an expression for A (t). Use a computer algebra system to perform the integration.