Question

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

How many points would you have to connect to form 99 regions?

Short answer

The number of points is 8.

Step-by-step solution

Step 1. Given

The number of regions is 99.

Step 2. To determine

We have to find the number of points.

Step 3. Calculation

The maximum numberof 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 99 regions, $f\left(n\right)=99$.

We solve the equation:

$\begin{array}{l}\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)=99\\ n^4-6n^3+23n^2-18n+24=24\left(99\right)\\ n^4-6n^3+23n^2-18n+24=2376\\ n^4-6n^3+23n^2-18n+24-2376=0\\ n^4-6n^3+23n^2-18n-2352=0\end{array}$

Using a graphing calculator, we solve this equation.

The two real zeros of the equation are -5.247 and 8.

n cannot be negative. So, n = 8.

The number of points is 8.