Calculus Please?

Question

If the rate of growth of cancer is modeled by: N'(t)=Ae^kt ; where A=rate at time= 0 days and K=a constant:
a) Suppose A=50 and at t=5 days, the cells are growing at a rate of 250 cells per day; find a formula for the number of cells after t days given that 300 cells are present at the beginning of this study.

b) Use this formula to calculate the number of cells that should be present after 12 days.

Answer 1

The number of cells, N(t) is just the integral of N'(t):

N(t) = A/k * e^(kt) + constant

Now you are given: N'(5) = 250 =50*e^(5k) so you can find k

5 = e^(5k) ---> ln(5) = 5k ---> k= ln(5)/5

And you were told that N(0) = 300, so

N(0)= A/k +constant = 300

A/k = 50*5/ln(5) = 155.33 so the constnat is

constant = 300 - 155.33 = 144.66

Then

N(t) = 155.33*e^(5t)+144.66

You can do part b