Chapter 13.3, Problem 14PPS

To determine

To calculate:

The probability that point chosen at the random lies in shaded region

Answer to Problem 14PPS

The probability of the point chosen at the random lies in shaded region is $0.4375$ .

Explanation of Solution

Given information:

  

Calculation:

In this given figure, if region $A$ contains region $B$ and point $E$ in region $A$ is chosen at random, then probability that the point $E$ is in region $B$ is area of region $A$ divide area of region $B$ .

  

The length of apothem of hexagon of the side length $a$ units is $\frac{a\sqrt{3}}{2}$ units.

Area of large hexagon:

  $\begin{array}{l}A=\frac{1}{2}Pa\\ A=\frac{1}{2}\times \left(6\times 8\right)\left(\frac{8\sqrt{3}}{2}\right)\\ A=24\left(\frac{8\sqrt{3}}{2}\right)\\ A=96\sqrt{3}\end{array}$

Area of small hexagons:

  $\begin{array}{l}A=4\left(\frac{1}{2}Pa\right)\\ A=4\left[\frac{1}{2}\times \left(6\times 3\right)\left(\frac{3\sqrt{3}}{2}\right)\right]\\ A=4\left[9\left(\frac{3\sqrt{3}}{2}\right)\right]\\ A=54\sqrt{3}\end{array}$

The area of shaded region is

  $96\sqrt{3}-54\sqrt{3}=42\sqrt{3}$

The probability is

  $\frac{42\sqrt{3}}{96\sqrt{3}}=\frac{7}{16}=0.4375$

The probability of the point chosen at the random lies in shaded region is $0.4375$ .