The perimeter of a polygon with sides of integer...

The perimeter of a polygon with sides of integer length is 45. If the smallest side of the polygon is 5 and the longest side of the polygon is 10, then the number of sides could be any number from?

A. 5 to 7
B. 5 to 8
C. 5 to 9
D. 6 to 8
E. 6 to 9

The OA is B.

I don't have clear this PS question.

I know that the perimeter is the sum of all the sides of the polygon, then
$$P=45=5+10+x_3+x_4+...+x_n$$
But, I don't know how can I get the x values to determine the number of sides.

I appreciate if any expert explain it for me. Thank you so much.

Hi AAPL,
Let's take a look at your question.

The perimeter of the polygon is 45 and 2 sides, the smallest and longest are 5 and 10.
Therefore the sum of the rest of the sides will be 45 - 10 - 5 = 30

Also, length of the remaining sides will be between the smallest and the longest side, i.e. between 5 and 10 inclusive.

Using the smallest length, the side lengths of the polygon could be
5 + 5 + 5 + 5 + 5 + 5 = 30
So the total sides of the polygon with all the remaining sides equal to 5 will be:
6 +2 = 8 sides

Using the longest length, the side lengths of the polygon could be
10 + 10 + 10 = 30
So the total sides of the polygon with all the remaining sides equal to 10 will be:
3 + 2 = 5 sides

Which shows that minimum number of sides of polygon is 5 and maximum number of sides could be 8.

Therefore, Option B is correct.

Hope it helps.
I am available if you'd like any follow up