SOLUTION: Help need with this question. Detailed workings would be very much appreciated, thanks. Joe decided to undergo an advanced typing course for 20 weeks and he progresses according

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Help need with this question. Detailed workings would be very much appreciated, thanks. Joe decided to undergo an advanced typing course for 20 weeks and he progresses according      Log On


   

Question 1045259: Help need with this question. Detailed workings would be very much appreciated, thanks.
Joe decided to undergo an advanced typing course for 20 weeks and he progresses according to the function
Q(t) = 130(1-e^(-0.05t) ) +65
where Q(t) is the number of words per minute Joe can type after t weeks in the course.
1) What is the initial typing speed of Joe in this course?
2) Find the typing speed of Joe halfway through the course.
3) In which week will Joe’s typing speed reach 100 words per minute?

Answer by advanced_Learner(501) About Me  (Show Source):
You can put this solution on YOUR website!
Q%28t%29 = 130%281-e%5E%28-0.05t%29+%29+%2B65
Q%28t%29 = 130%281-e%5E%28-0.05t%29+%29+%2B65
a.
initial means t=0
Q%28t=0%29 = 130%281-e%5E%28-0.05%2A0%29+%29+%2B65
remember x%5E0=1
Q%28t=0%29 = 130%281-1+%29+%2B65
Q%28t=0%29 = %28%28130%280%29+%29+%2B65%29
Q%28t=0%29 = 0%2B65
Q%28t=0%29 = 65
b.
halfway through 20 week course is 10 at t=10
Q%28t%29 = 130%281-e%5E%28-0.05t%29+%29+%2B65

Q%28t=10%29 = 130%281-e%5E%28-0.05%2A10%29+%29+%2B65
Q%28t=10%29 = 130%281-e%5E%28-0.5%29+%29+%2B65
Q%28t=10%29 = 130%281-0.60653%29+%29+%2B65
Q%28t=10%29 = 130%280.393%29+%29+%2B65
Q%28t=10%29 = %2851.15+%29+%2B65
Q%28t=10%29 = 116.15
Q%28t=0%29 = 116

c.
means when Q%28t%29=100
Q%28t%29 = 130%281-e%5E%28-0.05t%29+%29+%2B65
Q%28t%29 = 130%281-e%5E%28-0.05t%29+%29+%2B65
100+= 130%281-e%5E%28-0.05%2At%29+%29+%2B65
100+-65= 130%281-e%5E%28-0.05%2At%29+%29+
35=130%281-e%5E%28-0.05%2At%29+%29+
35/%28130%29=%281-e%5E%28-0.05%2At%29+%29+
7/%2826%29=%281-e%5E%28-0.05%2At%29+%29+
7/%2826%29-1=%28-e%5E%28-0.05%2At%29+%29+
7/%2826%29-26%2F26=%28-e%5E%28-0.05%2At%29+%29+
7-26/%2826%29=%28-e%5E%28-0.05%2At%29+%29+
-19/%2826%29=%28-e%5E%28-0.05%2At%29+%29+
cancel - sign(minus)
19/%2826%29=%28e%5E%28-0.05%2At%29+%29+

19/%2826%29=%28e%5E%28-0.05%2At%29%29
take ln of both sides
ln(19/%2826%29=ln%28e%5E%28-0.05%2At%29%29
-0.1366=ln%28e%5E%28-0.05%2At%29%29
-0.1366=(-0.05*t)*%28lne%29
remember lne=1
%28-0.1366%29=-0.05t%2A%281%29
%28-0.1366%29/-0.05=t
%286.27%29=t
approxitely t=6 weeks
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