Answer:
To find the probability that a randomly chosen point in the figure lies in the shaded region, we need to divide the area of shaded region by the total area of the figure.
Question 36
Total Area
Area of a rectangle = width × length = 8 × 12 = 96 units²
Shaded Region Area
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
Probability
= Shaded region area ÷ total area
= 24π ÷ 96
= 0.7853981634...
= 78.5% (3 s.f.)
Question 37
Total Area
Area of a rectangle = width × length = 16 × 8 = 128 units²
Shaded Region Area
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
Probability
= Shaded region area ÷ total area
= 88 ÷ 128
= 0.6875
= 68.8% (3 s.f.)
Question 38
Total Area
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
Shaded Region Area
= Area of a circle with radius (4 + 2) - area of circle with radius 2
= π(6)² - π(2)²
= 32π units²
Probability
= Shaded region area ÷ total area
= 32π ÷ 100π
= 0.32
= 32%
