A rectangle has vertices at the points (9, 6), (9, -9), (-1, -9), and (-1, 6) on the coordinate plane. A circle of radius 5 lies within this rectangle. What is the probability that a random point (x, y) that lies within the rectangle also lies within the circle?
A. 1 – π/6 B. 1/2 C. 25/15π D. π/4 E. π/6
(E) Let's determine the area for the rectangle that has vertices at the points indicated above. A sketch will help you visualize the rectangle.
Its length is the difference between the y coordinates: 6 – (-9) = 15.
Its width is the difference between x coordinates: 9 – (-1) = 10.
The area of the rectangle = 10 × 15 = 150.
The circle, which has a radius of 5, has an area of: π(5²) = 25π.
The question is asking us to find the probability that a point within the rectangle will also lie within the circle. This probability is the area of the circle divided by the area of the rectangle.
Probability = 25π/150 = π/6. The correct answer is choice (E).
Could you, please, draw a sketch for this question?
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