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Psychologists in learning theory study learning curves, the graphs of the "performance function" P=P(t) of someone learning a skill as a function of the training time t. If M represents the maximal level of performance, it is noted that learning is at first rapid, and then it tapers off (the rate of learning decreases) as P(t) approaches M. The differential equation below describes this situation. 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)

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    You just need to solve a first-order linear ODE, which can be solved by standard methods. – 2017-02-14
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    I am not even sure how to set this up – 2017-02-14

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There you go: $$\frac{dP}{dt}=k(M-P)$$ $$\frac{dP}{dt}+kP=kM$$ $$e^{kt}\frac{dP}{dt}+ke^{kt}P=ke^{kt}M$$ $$\frac{d(e^{kt}P)}{dt}=d(e^{kt}M)$$ $$e^{kt}P=e^{kt}M+C$$ For $t=0$: $$P(0)=0.1M=M+C$$ $$C=-0.9M$$ Hence $$P(t)=M-0.9Me^{-kt}P$$ Now you are able to find the required $t$ to reach the desired $P$. After simple calculations we compute $$t=\frac{\ln9}k$$

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    Is t=ln(9)/k the final answer or do you have to find what k is as well. – 2017-02-14
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    @JonJones k is given and it is equal to 0.05 (as you said in your question) . Dont just copy the answer, make sure to understand it. – 2017-02-14