Chapter 13.3, Problem 28PPS

(a)

To determine

To calculate:

The probability that point chosen at random lies inside the circle

Answer to Problem 28PPS

The probability of point chosen at random lies inside the circle is $0.13$ .

Explanation of Solution

Given information:

  

Calculation:

The coordinate grid contains $100$ square units. In the grid is circle with the radius of $2$ units, a square with the side length $2\sqrt{2}$ units and trapezoid with the base lengths $2$ units and $4$ units and the height of $3$ units.

The probability that point randomly chosen lies within the circle is equal to the area of circle divided by the total area of grid.

  $\begin{array}{l}P\left(\text{X within circle}\right)=\frac{\text{area}\text{of}\text{circle}}{\text{area}\text{of}\text{grid}}\\ P\left(\text{X within circle}\right)=\frac{\pi {\left(2\right)}^2{\text{units}}^2}{100{\text{units}}^2}\\ P\left(\text{X within circle}\right)=\frac{\pi }{25}\\ P\left(\text{X within circle}\right)=0.13\end{array}$

The probability of point chosen at random lies inside the circle is $0.13$ .

(b)

To determine

To calculate:

The probability that point inside the trapezoid

Answer to Problem 28PPS

The probability that chosen randomly plot is used to grow soybeans is $0.16$ .

Explanation of Solution

Given information:

  

Calculation:

The probability that point randomly chosen lies within the circle is equal to the area of circle divided by the total area of grid.

  $\begin{array}{l}P\left(\text{X within trapezoid}\right)=\frac{\text{area}\text{of}\text{trapezoid}}{\text{area}\text{of}\text{grid}}\\ P\left(\text{X within trapezoid}\right)=\frac{\frac{1}{2}\left(3\right)\left(2+4\right){\text{units}}^2}{100{\text{units}}^2}\\ P\left(\text{X within trapezoid}\right)=\frac{9}{100}\\ P\left(\text{X within trapezoid}\right)=0.09\end{array}$

The probability of point chosen at random lies inside the trapezoid is $0.09$ .

(c)

To determine

To calculate:

The probability that point inside the trapezoid, square or circle

Answer to Problem 28PPS

The probability that chosen randomly plot is used to grow soybeans is $0.16$ .

Explanation of Solution

Given information:

  

Calculation:

The probability that point randomly chosen lies within the circle is equal to the area of circle divided by the total area of grid.

  $\begin{array}{l}P\left(\text{X within any figure}\right)=\frac{\text{area}\text{of}\text{three figures}}{\text{area}\text{of}\text{grid}}\\ P\left(\text{X within any figure}\right)=\frac{4\pi +\frac{1}{2}\left(3\right)\left(2+4\right)\text{+}\left(2\sqrt{2}\right)\left(2\sqrt{2}\right){\text{units}}^2}{100{\text{units}}^2}\\ P\left(\text{X within any figure}\right)=\frac{30}{100}\\ P\left(\text{X within any figure}\right)=0.30\end{array}$

The probability of point chosen at random lies inside the trapezoid is $0.30$ .