Expert: Alon Mandes - 2/14/2009

Question
1The number of bus riders from the suburbs to downtown per day is represented by 1200(1.50-x), where x is the fare in dollars. What fare will maximize the total revenue?

#2 A large social part y follow the mathematical progression
N(t)=30t-t^2, where t is the time in minutes since the party began and N is the number of separate conversation occuring. At what time in a party do the most converstaions occur? what is the maximim number of interactions?

#3A new cottage is built across the river and 300 m downstream from the nearest telephone station. The river is 120 m wide. In order to wire the cottage for phone service, wire will be laid across the river underwater, and along the edge of the river above ground. The cost to laywire under water is $15 per m and the cost to lay wire above ground is 10 per m. How much wire should be laid under water to minimize the cost?

I am very stuck with these, because the book does not do a good job of explaning and my deadline is fast approaching, help is much appreciated

Answer
#2
N'(t)=30-2t. N'(t)=0 -> t=15 min. N(15)=30*15-15*15=225.

#3
Coast=x*15+y*10
According to Pythagoras (300-y)^2+120^2=x^2. Thus,
Coast(y)=15*sqrt[(300-y)^2+120^2]+10y .
Coast'(t)=-15*(300-y)/{sqrt[(300-y)^2+120^2]} +10
Coast'(t)=0 -> (2/3)=(300-y)/{sqrt[(300-y)^2+120^2]} ->
(2/3)*{sqrt[(300-y)^2+120^2]}=300-y
(4/9)[(300-y)^2+120^2]=(300-y)^2
I'll leave it to you as an exercise to continue & find y.

Alon.