SOLUTION: In each of the following, determine the number of sides of a regular polygon with the stated property. If such a regular polygon does not exist, explain why. b. Each exterior

Algebra ->  Polygons -> SOLUTION: In each of the following, determine the number of sides of a regular polygon with the stated property. If such a regular polygon does not exist, explain why. b. Each exterior       Log On


   

Question 172385This question is from textbook
: In each of the following, determine the number of sides of a regular polygon with the stated property. If such a regular polygon does not exist, explain why.
b. Each exterior angle measure 25 degrees
d. The total number of diagonals is 4860 This question is from textbook

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
b)since the sum of the exterior angles must equal 360...we have 360/25=14.4....therefore there is no such polygon since this is not a positive integer
:
d)the formula for diagonals is as follows:d= n/2 (n-3)
where d is the number of diagonals and n is the number of sides in the polygon
: the answer needs to be a positive integer
4860=n/2(n-3)
4860=%28n%5E2-3n%29%2F2
:
9720=n%5E2-3n
:
n%5E2-3n-9720=0
this does not produce a positive integer solution therefore this polygon does not exist
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation an%5E2%2Bbn%2Bc=0 (in our case 1n%5E2%2B-3n%2B-9720+=+0) has the following solutons:

n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-3%29%5E2-4%2A1%2A-9720=38889.

Discriminant d=38889 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+38889+%29%29%2F2%5Ca.

n%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+38889+%29%29%2F2%5C1+=+100.101470577269
n%5B2%5D+=+%28-%28-3%29-sqrt%28+38889+%29%29%2F2%5C1+=+-97.1014705772688

Quadratic expression 1n%5E2%2B-3n%2B-9720 can be factored:
1n%5E2%2B-3n%2B-9720+=+1%28n-100.101470577269%29%2A%28n--97.1014705772688%29
Again, the answer is: 100.101470577269, -97.1014705772688. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-3%2Ax%2B-9720+%29