Expert: Sherry Wallin - 12/6/2008

Question
A newspaper is launching a new advertising campaign in order to increase the number of daily
subscribers. The newspaper currently (t = 0) has 26,000 daily subscribers and management expects
that number, S(t), to grow at the rate of S'(t) = 80t1/2 subscribers per day, where t is the number of
days since the campaign began. How long (to the nearest day) should the campaign last if the
newspaper wants the number of daily subscribers to grow to 49,000?

Thank you for your help understanding this!

Answer
You are given an ordered pair (time, subscribers)= (t,s) and you are also given that you have 26,000 subscribers on the starting day of the campaign, this translates into a point (0, 26000). You are also given that you want to know when the number of subscribers is 49000, so you are looking for (t,490000). You can calculate the slope and you are given the slope(that is what the first derivative) so set them equal to each other.
slope = m = (49000-26000)/(t-0)=23000/t
so 23000/t = 80t^1/2 so 23000 = 80t^3/2 and 23000/80 = t^3/2
Now raise both sides of this equation to the 2/3 power
(23000/80)^2/3 = t which is 43.56 or approximately 44 days to raise the number of subscribers to 49000.
Note you can check this by calculating the number of new subscribers each day. One day 1 there would be 80*1^1/2 = 80, on day 2 there would be 80*2^1/2 about 113, and on the 3rd day 80*3^1/2 or about 139 and add all these up for the 44 days and you will get slightly more than 49000 since we rounded the 43.56 up to 44.

I hope this is clear and helps.
Math Prof