Math 131		Assignment 29 Answer

2. Do p. 204 #27.  Explain your work.  
   Include a sign of the derivative chart.

I will use the steps on page 200.

Step 0. I want to and I can.
Step 1. Find minimum and maximum revenue.  Revenue equals the product of
the number of chairs sold and the price of the chair.
I do not know the number of chairs sold.  I do know that price depends on
the number of chairs sold.  So let c be the number of chairs sold.  The
domain of c is 0, 1, 2, 3, 4, ... 400.  To use calculus I will assume c
is a real variable rather than an integer variable.  I do need to round
off if calculus provides a non integer solution.

Step 2.  I do not need a sketch.

Step 3. I need a formula for revenue = c * price.
Now price depends on the number of chairs sold.  If c <= 300, the price of a
chair is fixed at $90.  If c is between 300 and 400 then the price of a
chair decreases by a quarter for each chair over 300 ordered.  What does
this mean?  A few examples will help me find a formula for price that depends
on c.  If c = 301, the price is $90 - 0.25.  If c = 302, the price is 
$90 - 2 * 0.25.  If c = 303, the price is $90 - 3 * 0.25.  
So if c is between 300 and 400 the price is $90 - ( c - 300 ) * 0.25.
So revenue = c * price = c * 90 if 0 <= c <= 300
		       = c * ( 90 - ( c - 300 ) * 0.25 ) if 300 < c <= 400.
I will do arithmetic on revenue
r = 90 c if 0 <= c <= 300
  = 90 c - 0.25 c^2 + 75 c = 165 c - 0.25 c^2 if 300 < c <= 400.

Step 4.
Find r' and solve r' = 0.
r' = 90 if 0 <= c <= 300
   = 165 - 0.5 c if 300 < c <= 400.
Solve r' = 0.
r' can be zero Only when c is between 300 and 400.
Solve 0 = 165 - 0.5 c
Add 0.5 c
0.5 c = 165
Multiply by 2
c = 330.

Make a sign of r' chart
c	320 	330 	340
r'	 +	 0	 -
since r'( 320 ) = 165 - 0.5 * 320 = 165 - 160 = 5
and   r'( 340 ) = 165 - 0.5 * 340 = 165 - 170 = -5.
So c = 330 is a maximum where revenue is 
r( 330 ) = 165 * 330 - 0.25 * 330^2 = 27225.
r' does not exist at 300.  So I need to check revenue for 300 chairs.
r( 300 ) = 90 * 300 = 27000.
So the maximum revenue is $27225 which occurs for an order of 330 chairs.
The minimum revenue is $0 which occurs for an order of 0 chairs.