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McGraw Hill Integrated II, 2012

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McGraw Hill Integrated II, 2012 View details

3. Geometric Probability

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Exercise 35 Page 904

To find the area of the shaded region, we can subtract the area of the unshaded region from the area of the figure.

≈ 14.3 %

Practice makes perfect

To find the probability that a randomly chosen point would lie in the shaded area of the figure, we will use geometric probability. The probability that the point lies in the shaded area is the ratio of the area of the shaded part of the figure to the area of the entire figure.

Finding the Area of the Figure

Let's take a closer look at the given figure.

As we can see, our figure is a rectangle. The middle part contains a circle with radius r. The diameter of this circle, and also the length of this part of the figure, is 2r. This is also the width of the entire figure. The left and right parts of our figure each contain a semicircle with radius r, so the length of both of these parts is also 2r. This means that the length of the figure is 6r.

Knowing the dimensions of the figure, we can calculate its area. Area of the Figure = 2r * 6r ⇕ Area of the Figure = 12r^2

Finding the Area of the Shaded Region

To find the area of the shaded region, we will subtract the area of the unshaded region from the area of the figure. Let's take a closer look at the unshaded region.

The unshaded region consists of 2 halfcircles with radius r, a circle with radius r, and two rectangles with length 2r and width r. We can calculate the area of each of these figures.

Figure	Area of the Figure
Halfcircle	H = 1/2π r^2
Circle	C = π r^2
Rectangle	R = 2r* r

Let's now calculate the area of the unshaded region!

Area of the Unshaded Region = 2H + C+ 2R

SubstituteValues

Substitute values

Area of the Unshaded Region = 2( 1/2π r^2) + π r^2+ 2( 2r* r)

Multiply

Area of the Unshaded Region = π r^2 + π r^2+ 4r^2

Add terms

Area of the Unshaded Region = 2π r^2+ 4r^2

Knowing that the area of the unshaded region is 2π r^2+ 4r^2, we can calculate the area of the shaded region.

Area of the Shaded Region = Area of the Figure - Area of the Unshaded Region

Area of the Figure= 12r^2, Area of the Unshaded Region= 2π r^2+ 4r^2

Area of the Shaded Region = 12r^2 - ( 2π r^2+ 4r^2)

Distribute (- 1)

Area of the Shaded Region = 12r^2 - 2π r^2- 4r^2

Subtract term

Area of the Shaded Region = 8r^2 - 2π r^2

Factor out 2r^2

Area of the Shaded Region = 2r^2(4 - π)

Calculating the Probability

Finally, we can calculate the probability that a point chosen at random would lie in the shaded region of the figure.

P(point is in the shaded region) = Area of the shaded region/Area of the figure

Area of shaded region= 2r^2(4 - π), Area of figure= 12r^2

P(point is in the shaded region) = 2r^2(4 - π)/12r^2

a/b=.a /2r^2./.b /2r^2.

P(point is in the shaded region) = 4 - π/6

Use a calculator

P(point is in the shaded region) = 0.858407.../6

Calculate quotient

P(point is in the shaded region) = 0.143068...

Convert to percent

P(point is in the shaded region) = 14.3068... %

Round to 1 decimal place(s)

P(point is in the shaded region) ≈ 14.3 %

The probability that a point chosen at random would lie in the shaded area of the figure is approximately 14.3 %.

Subchapter links

Exercises p.902-905