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Question

A polygon with n sides has D​ diagonals, where D is given by the function

D(n)=n(n−3)2.

Find the number of sides n if 104≤D≤560.

The number of sides is​ {n | _ ≤n≤ _ }.

Answer 1

Take the function D(n) = n(n−3)/2 and plug in D = 104 and then D = 560. Solve for "n" in both cases. That will give the the range of values for "n":

104 = n(n−3)/2

104(2) = n² - 3n

n² - 3n - 208 = 0

(n + 13)(n - 16) = 0

So n = -13 and n = 16

Since you cannot have a polynomial of side length -13, you ignore that solution.

Repeat for

560 = n(n−3)/2 and n = 16 and whatever you get from solving this equation will be your values that go in the blanks.