Calculus 2

Question

The graphs of the "performance function" P= P(t) of someone learning a skill as a function of the training time t. Suppose that for a specific learning activity, it is determined that P(0) = 0.1M and k= 0.05 for t measured in hours.

How long does it take to reach 90% of the maximal level of performance M?

(dP/dt)= k(M-P)

Answer 1

This is separable.

dP/(M-P)=kdt

 

do you know how to integrate?

 

you should get something like

-ln|M-P| = kt + C

 

use initial conditions P(0) = 0.1M

thus -ln|M-0.1M|=-ln|0.9M| = -ln(0.9M) = C assuming M is > 0.

 

we want to know when P(t) = 0.9M

plug in

-ln|M-0.9M|=-ln(0.1M)=kt-ln(0.9M)

add and remember log properties you have

t = ln(9)/k and since k = 1/20 you have

t = 20ln(9)

 

best - david