Expert: Scotto - 2/17/2009

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
1. Take 1200(1.50-x) times x.  

Take the derivative, set it equal to 0, solve for x.
This is the resulting fair.  It is a maximum for the function
since the second derivative of the function is negative.


2. Take the derivative, set it equal to 0, and solve for t.
Call the answer t0.

This is the time in the party
at which most of the interactions occur.

The maximum number of interactions is N(t0).


3. The distance down river is 300 and the width is 120.
Let 300-x be how far the wire runs alng the bank.
The distance it runs downstream below the water is x.
The distance across the river is runs is 120.

This means the total distance across the river is √(120²+x²) =
√(14,400+x²).  Since the cost underwater is 15 and the cost on
the shoreline is 10, the total cost of setting this wire is
C(x) = 15√(14,400+x²) + 10x.

Take the derivative { which involves a chain rule },
set the equation equal to 0, and solve for x.

Note that the derivative { not yet reduced to simplest form } is
(15(0.5)/√(14,400+x²))2x + 10.
Move the 2x out front { since it's in the numerator },
cancel the 2 and the 0.5 { the product is 1 }, and solve for x.

Note that x was the distance on the bank the cable was underwater.
The answer is then √(14,400+x²).