A polygon with \( n \) sides can have diagonals that are calculated using the formula:
\[ \text{Number of diagonals} = \frac{n(n-3)}{2} \]
We know the polygon has 105 diagonals. Thus, we set up the equation:
\[ \frac{n(n-3)}{2} = 105 \]
Solve for \( n \) by first multiplying both sides by 2 to clear the fraction:
\[ n(n-3) = 210 \]
Now, expand and rearrange to form a standard quadratic equation:
\[ n^2 - 3n - 210 = 0 \]
Next, we factorize the quadratic equation:
\[ (n-15)(n+14) = 0 \]
The solutions for \( n \) are obtained by setting each factor to zero:
We disregard \( n = -14 \) as a polygon cannot have negative sides.
Therefore, the solution is: \( n = 15 \)
The polygon has 15 sides.
To determine the number of sides in a polygon given the total number of lines connecting its vertices, we follow these steps:
1. Problem Analysis:
The number of lines connecting any two vertices (sides and diagonals) of an n-sided polygon is given by the combination formula:
$ \binom{n}{2} = \frac{n(n-1)}{2} $
2. Given Condition:
We are told the total number of lines is 105:
$ \frac{n(n-1)}{2} = 105 $
3. Solving the Equation:
Multiply both sides by 2:
$ n(n-1) = 210 $
Rearrange to standard quadratic form:
$ n^2 - n - 210 = 0 $
4. Factoring the Quadratic:
Find factors of -210 that sum to -1:
$ n^2 - 15n + 14n - 210 = 0 $
Factor by grouping:
$ (n - 15)(n + 14) = 0 $
5. Valid Solution:
The solutions are n = 15 or n = -14.
Since a polygon can't have negative sides, we take:
$ n = 15 $
Final Answer:
The polygon has $15$ sides.
A solid cylinder of mass 2 kg and radius 0.2 m is rotating about its own axis without friction with angular velocity 5 rad/s. A particle of mass 1 kg moving with a velocity of 5 m/s strikes the cylinder and sticks to it as shown in figure.
The angular velocity of the system after the particle sticks to it will be: