Question

Under certain conditions, the number of diseased cells N(t) at time t increases at a rate N'(t)=Ae^kt, where A is the rate of increase at time 0 (in cells per day) and k is a constant. A. Suppose

A = 40 and at 6 days the cells are growing at a rate of 120 per day. Find a formula for the number of cells after t days, given that 200 cells are present at t=0.

B. Use your answer from part a to find the number of cells present after 16 days.

Answer 1

Step 1)

As given we have,

we have A = 40 hence we can write,

At 6 days cells are growing at a rate of 120 per day hence put t = 6 and N(t) = 120 we can write,

Hence we can write,

Hence,

integrate on both the side we can say that,

--------------------------------------------------------1)

At t = 0 we have 200 cells present hence put t = 0 and N = 200 in equation 1) we can write,

Put this value in equation 1) we can write,

we can write number of cells after t days is given by

-----------------------------------------------------2)

In decimals we can write,

Step 2)

we have,

we have to find the number of cells present after 16 days hence put t = 16 we can write,

we know that eln(a) = a hence,

rounding to nearest integer we can say that,

Hence we can say that number of cells present after 16 days is 4072