Q54. - Algebra 2

Question

PATTERNS For Exercises 53–55, use the diagrams below that show the maximum number of regions formed by connecting points on a circle.

Find the maximum number of regions formed by connecting 5 points of a circle.

Draw a diagram to verify your solution.

Step-by-Step Solution

Answer

For 5 points the maximum number of regions is 16.

Diagram is:

Step 1. Given

The diagrams below that show the maximum number of regions formed by connecting points on a circle. 

Step 2. To determine

We have to find the maximum number of regions formed by connecting 5 points of a circle.

Step 3. Calculation

From the diagram, 

The maximum number of regions formed by connecting n points of a circle can be described by the function:

$f\left(n\right)=\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)$

So, for 5 points, $n=5$.

So, the maximum number of regions is:

 $\begin{array}{l}f\left(n\right)=\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)\\ f\left(5\right)=\frac{1}{24}\left(5^4-6{\left(5\right)}^3+23{\left(5\right)}^2-18\left(5\right)+24\right)\\ f\left(5\right)=16\end{array}$

Diagram is: