Answer:
To find the probability that a randomly chosen point in the figure lies in the <u>shaded region</u>, we need to divide the area of shaded region by the total area of the figure.
<h3><u>Question 36</u></h3>
<u>Total Area</u>
Area of a rectangle = width × length = 8 × 12 = 96 units²
<u>Shaded Region Area</u>
This is made up of 6 congruent circles.
The radius of each circle is 1/6 of the length of the rectangle (or 1/4 of the width).
⇒ radius = 12/6 = 2 units
Area of a circle = πr² = π(2)² = 4π units²
⇒ Shaded region area = 6 circles = 6 × 4π = 24π units²
<u>Probability</u>
= Shaded region area ÷ total area
= 24π ÷ 96
= 0.7853981634...
= 78.5% (3 s.f.)
<h3><u>Question 37</u></h3>
<u>Total Area</u>
Area of a rectangle = width × length = 16 × 8 = 128 units²
<u>Shaded Region Area</u>
The easiest way to calculate this is to subtract the un-shaded areas from the total area:
⇒ Shaded region area = 128 - 2(2 · 10) = 88 units²
<u>Probability</u>
= Shaded region area ÷ total area
= 88 ÷ 128
= 0.6875
= 68.8% (3 s.f.)
<h3><u>Question 38</u></h3>
<u>Total Area</u>
The radius of the largest circle is the sum of the individual given radii.
⇒ Area of a circle = πr² = π(4 + 4 + 2)² = 100π units²
<u>Shaded Region Area</u>
= Area of a circle with radius (4 + 2) - area of circle with radius 2
= π(6)² - π(2)²
= 32π units²
<u>Probability</u>
= Shaded region area ÷ total area
= 32π ÷ 100π
= 0.32
= 32%
