Question

find the probability that a randomly chosen point in each figure lies in the shaded region. hint: p(shaded region) = \\(\frac{\text{area of shaded region}}{\text{area of entire figure}}\\) figures: a) square (side 8 in) with two semicircles (white); b) annulus (inner diameter 8 in, outer diameter 10 in); c) rectangle (25 m × 22 m) with a white triangle

Explanation:

Part (a)

Step 1: Find area of square

The square has side length \( 8 \) in. Area of square \( A_{square} = s^2 = 8^2 = 64 \) in².

Step 2: Find area of two semicircles (which make a circle)

The diameter of each semicircle is \( 8 \) in, so radius \( r = \frac{8}{2}=4 \) in. Area of a circle \( A_{circle}=\pi r^2=\pi(4)^2 = 16\pi \) in².

Step 3: Find area of shaded region

Shaded area \( A_{shaded}=A_{square}-A_{circle}=64 - 16\pi \) in².

Step 4: Find probability

Probability \( P=\frac{A_{shaded}}{A_{square}}=\frac{64 - 16\pi}{64}=\frac{4-\pi}{4}\approx\frac{4 - 3.14}{4}=\frac{0.86}{4} = 0.215 \) (or in terms of \( \pi \), \( 1-\frac{\pi}{4} \)).

Step 1: Find radii of circles

Outer circle diameter is \( 10 \) in, so radius \( R=\frac{10}{2} = 5 \) in. Inner circle diameter is \( 8 \) in, so radius \( r=\frac{8}{2}=4 \) in.

Step 2: Find area of outer circle

\( A_{outer}=\pi R^2=\pi(5)^2 = 25\pi \) in².

Step 3: Find area of inner circle

\( A_{inner}=\pi r^2=\pi(4)^2 = 16\pi \) in².

Step 4: Find area of shaded region (annulus)

\( A_{shaded}=A_{outer}-A_{inner}=25\pi - 16\pi=9\pi \) in².

Step 5: Find probability

Probability \( P=\frac{A_{shaded}}{A_{outer}}=\frac{9\pi}{25\pi}=\frac{9}{25}=0.36 \).

Step 1: Find area of rectangle

Rectangle has length \( 25 \) m and width \( 22 \) m. Area \( A_{rectangle}=l\times w=25\times22 = 550 \) m².

Step 2: Find area of unshaded triangle

The triangle has base \( 25 \) m and height \( 22 \) m. Area of triangle \( A_{triangle}=\frac{1}{2}\times b\times h=\frac{1}{2}\times25\times22 = 275 \) m².

Step 3: Find area of shaded region

\( A_{shaded}=A_{rectangle}-A_{triangle}=550 - 275 = 275 \) m².

Step 4: Find probability

Probability \( P=\frac{A_{shaded}}{A_{rectangle}}=\frac{275}{550}=\frac{1}{2}=0.5 \).

Answer:

\( 1-\frac{\pi}{4} \) (or approximately \( 0.215 \))

Part (b)