An epidemic hits a college campus, beginning with 7 cases on the first day...?

Question

The rate of growth of the epidemic (new cases per day) is given by the function r(t) = 30e^0.05t.

a) Find a formula for the total number of cases of flu in the first t days.

b) Use that formula to find the total number of cases in the first 15 days.

Show all work for both part a and b.

Thank you so much in advance for your help with this. I just cannot figure it out!

Answer 1

If r(t) is the rate, and we want the number of cases, we need to integrate. The integral of r(t) is 600e^0.05t + C. We know that there are 7 cases on the first day, so 7 = [600e^0.05(1) + C]. Thus we have C = 7 - 600e^0.05.

So the number of cases on day t, call it f(t), is given by:
f(t) = 600e^0.05t + 7 - 600e^0.05
f(t) = 600e^(0.05(t+1)) + 7

Then there are f(15) cases in the first 15 days. Just plug in 15.

I did the algebra part admittedly too fast so you'll have to check that over, but the calculus concepts are right.

Answer 2

The total cases after t days is given by
Int 30e^0.05t dt = 30/0.05 e^0.05 t +C
At t=0 this should be 7 so 600+C=7
C=-593
Total case in the first t days N(t) = 600 e^0.05t -593
for t=15
N(15) = 677 rounded to the nearest integer