A farmer wishes to enclose the maximum possible area with 100 meters of fence. The pasture is bordered by a straight cliff, which may be used as part of the fence. What is the maximum area that can be enclosed?

[*] this post was modified to correct formatting problems in the formulas and correct the Basic program to use radians for the SIN() function...

 

Area of a segment is given as:

Area = 1/2•R^2•(PI/180•THETA-Sin(THETA) in degrees

arc length is given as:

Arc Length = (THETA•PI)/180•R  in degrees

100 = (THETA•PI•R)/180
18000 = THETA•PI•R
r = 18000/(THETA•PI)

If I plug 18000/(THETA•PI) back intot he original formula for the area of a segment in place of R I get:

Area = 1/2•(18000/(THETA•PI))^2•(PI/180•THETA-Sin(THETA) <- assuming that SIN() uses degrees...

I then use the following program to check each degree from 1 to 180 to see which gives the maximum area for the segment:

Dim I, Degrees As Integer
Dim Area, TopArea, Pi As Double
 
Pi = 3.1415926
Degrees = 0
TopArea = 0
 
For I = 1 To 180
    Area = (1 / 2) * (18000 / (I * Pi)) ^ 2 * (Pi / 180 * I - Sin(Pi * I / 180))
    If Area > TopArea Then
        TopArea = Area
        Degrees = I
    End If
Next

 
Text1.Text = Degrees
Text2.Text = TopArea

This gives me the maximum area of 848006 m² and a value of degrees of 4. [*]

What did I do wrong?  Did I forget to convert between degrees and radians somewhere?[*]

 

[*]  The corrected program returns a value of 180° for Theta and a total area for the segment as 1591.5494...

Edited on June 18, 2004, 1:14 pm

Edited on June 18, 2004, 1:34 pm

Posted by Erik O. on 2004-06-18 10:51:31