3. The more you study for a certain exam, the better your performance will be on

Question

3. The more you study for a certain exam, the better your performance will be on it. You can get as close as you want to a perfect score by studying long enough. We can model this situation by a rational function of the form ax + b f(x) = where x is the amount of hours of study, and f(x) is your score on the exam. To find our model you need to find the values of a, b and d for this function. (a) The idea that you will get as close as you like to a perfect score, in this context translates to the sentence f(x) has a horizontal asymptote at y = 100, Use this information to find the value of a. (b) Assume that if you study for 10 hours, your score is 60%, but if you study for 20 hours, your score is 90%. Use this information to find the values of b and d. (c) Write the function f(x), with the values of a, b and d you found above. (d) Use this function to find how many hours you need to study in order to get a score of 95% on the exam.

Explanation / Answer

a. SInce the degree of numerator and denominator are equal. So horizontal asymptote y is the ratio of numerator's leading coefficient and denominator's leading coefficient

y=a/1

y=a

And it is given that y=100 is the horizontal asymptote. It means a=100.

b.0.6= (100*10 +b)/(10+d)

6+0.6d=1000+b

0.6d -b=994

0.9=(100*20+b)/(20+d)

1.8+0.9d= 2000+b

0.9d-b=1998.2

b=1014.4 , d=3347.33

d. f(x) =(100x +1014.4)/(x+3347.33)

0.95 =(100x +1014.4)/(x+3347.33)

x=21.86 hours

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