Here we want to solve differential equations, we will see that the general solution is:
![y = A*e^{-t} + 100](https://tex.z-dn.net/?f=y%20%3D%20A%2Ae%5E%7B-t%7D%20%2B%20100)
We want to solve the differential equation:
![\frac{dy}{dt} = 100 - y](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdt%7D%20%20%3D%20100%20-%20y)
From this is pretty clear that y is an exponential function, with an exponent of -1*t.
We can write it generally as:
![y = A*e^{-t} + B\\\\\frac{dy}{dt} = -A*e^{-t}](https://tex.z-dn.net/?f=y%20%3D%20A%2Ae%5E%7B-t%7D%20%2B%20B%5C%5C%5C%5C%5Cfrac%7Bdy%7D%7Bdt%7D%20%20%3D%20-A%2Ae%5E%7B-t%7D)
Then if we set B = 100 we get:
![y = A*e^{-t} + 100\\\\\frac{dy}{dt} = -A*e^{-t} \\\\\frac{dy}{dt} = -A*e^{-t} - 100 + 100 = -y + 100](https://tex.z-dn.net/?f=y%20%3D%20A%2Ae%5E%7B-t%7D%20%2B%20100%5C%5C%5C%5C%5Cfrac%7Bdy%7D%7Bdt%7D%20%20%3D%20-A%2Ae%5E%7B-t%7D%20%5C%5C%5C%5C%5Cfrac%7Bdy%7D%7Bdt%7D%20%20%3D%20-A%2Ae%5E%7B-t%7D%20-%20100%20%2B%20100%20%3D%20-y%20%2B%20100)
So we just found the general form of the function.
Now we have two cases:
A) y(0) = 35
![y(0) = A*e^{-0} + 100 = 35\\= A + 100 = 35\\\\A = 35 - 100 = -65](https://tex.z-dn.net/?f=y%280%29%20%3D%20A%2Ae%5E%7B-0%7D%20%2B%20100%20%20%3D%2035%5C%5C%3D%20%20A%20%2B%20100%20%3D%2035%5C%5C%5C%5CA%20%3D%2035%20-%20100%20%3D%20-65)
In this case, the function is:
![y = -65*e^{-t} + 100](https://tex.z-dn.net/?f=y%20%3D%20-65%2Ae%5E%7B-t%7D%20%2B%20100)
B) y(0) = 125
![125 = A*e^0 + 100\\\\\125 - 100 = A\\\\25 = A](https://tex.z-dn.net/?f=125%20%3D%20A%2Ae%5E0%20%2B%20100%5C%5C%5C%5C%5C125%20-%20100%20%3D%20A%5C%5C%5C%5C25%20%3D%20A)
In this case, the function is:
![y = 25*e^{-t} + 100](https://tex.z-dn.net/?f=y%20%3D%2025%2Ae%5E%7B-t%7D%20%2B%20100)
Now we want to see which one of the two can represent how a person learns. Just look at the graph below:
The green line is the one for y(0) = 35, and the blue one is for y(0) = 125.
Notice that for small values of t, the blue function is really large, thus it can't really model how a person learns (is larger for smaller values of t than for larger values).
So y(0) = 35 represents better how a <em>person can learn</em> (but not exactly, because you can see that it eventually becomes almost constant, which is something that really does not happen) so the correct option is <u>D: none of the above.</u>
If you want to learn more, you can read:
brainly.com/question/353770