An instructor has determined that students' grads on the final exam depend on the amount of time, t, they spend studying for it and can be modeled by the formula g(t)=(21t+10)/(20t+45). Note that the function's output is a fraction or decimal which should then be converted to a percent to give the student's grade on the exam.
-If the student were able to study as long as s/he desired, what score would s/he approach (but never actually reach)?
This is how I tried to solve this question:
Eliminate 10 and 45
g(t)---> 21/20=1.05
So would the answer be that the student would approach 105%? (but never actually reached)
Any help is greatly appreciated!
That is correct. You can see that at t=35, g(t) = 1
So, a student will likely get 100% if he spends 35 hours studying. I guess it's unlikely that more time than that will be spent...
To find the score that the student would approach (but never actually reach) as they study for an infinite amount of time, we can analyze the formula g(t) = (21t + 10)/(20t + 45).
As you correctly observed, we can simplify the formula by eliminating the constants 10 and 45:
g(t) = (21t + 10)/(20t + 45)
= (21/20)(t + 10/21)/(t + 45/20).
Now, let's analyze what happens as t (the amount of time spent studying) approaches infinity.
As t becomes larger and larger, the terms with the smallest powers (10/21 and 45/20) become relatively negligible compared to the term with the largest power (t). Therefore, we can neglect these terms as t approaches infinity.
The simplified expression then becomes:
g(t) ≈ (21/20)(t/t) = 21/20
This means that as the student studies for an infinite amount of time, their score will approach 21/20 or 105%.
So, you are correct. The answer is that the student would approach a score of 105%, but they would never actually reach it.