A company with a large customer base has a call center that receives thousands of calls a day. After studying the data that represents how long callers wait for assistance, they find that the function \(p(t)=0.25e^{-0.25t}\) models the time customers wait in the following way: the fraction of customers who wait between \(t = a\) and \(t = b\) minutes is given by
\begin{equation*}
\int_a^b p(t) \, dt.
\end{equation*}
Use this information to answer the following questions.
(a) The fraction of callers who wait between 5 and 10 minutes is .
(b) The fraction of callers who wait between 10 and 20 minutes is .
(c) Next, let us study the fraction who wait up to a certain number of minutes:
(i) What is the fraction of callers who wait between 0 and 5 minutes?
Answer:
(ii) What is the fraction of callers who wait between 0 and 10 minutes?
Answer:
(iii) Between 0 and 15 minutes?
Answer:
(iv) Between 0 and 20?
Answer:
(d) Let \(F(b)\) represent the fraction of callers who wait between \(0\) and \(b\) minutes. Find a formula for \(F(b)\) that involves a definite integral.
\begin{equation*}
F(b)=\int_c^d f(t) \, dt
\end{equation*}
where \(c=\), \(d=\), and \(f(t)=\).
Now evaluate that integral using the First FTC to find a formula for \(F(b)\) that does not involve a definite integral (but will involve \(b\) ).
\(F(b)=\)
(e) Calculate \(\displaystyle \lim_{b \to \infty} F(b) =\)
Why does that answer make sense?
Because no one waits more than an hour.
Because every customer waits between 0 and infinity minutes.
Because one customer is helped at a time.
none of the above