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 Post subject: GMAT Coordinate Geometry
PostPosted: Thu May 13, 2010 2:20 pm 
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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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 Post subject: Re: math: x,y-plane and probability
PostPosted: Thu May 13, 2010 2:23 pm 
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Here is the sketch of how rectangle is situated:
Image

Note, that we do not need to know how circle is aligned within the rectangle, because for any alignment the probability would be the same.
Image
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