Three circles have the same center. Their radii measure 1, 2, and 3 inches, respectively. If a point is chosen at random in the interior of the largest circle, what is the probability that the point is also in the interior of the smallest circle?
Question
Three circles have the same center. Their radii measure 1, 2, and 3 inches, respectively. If a point is chosen at random in the interior of the largest circle, what is the probability that the point is also in the interior of the smallest circle?
Solution
The probability of a point chosen at random being in the interior of the smallest circle can be found by comparing the areas of the smallest and largest circles.
Step 1: Calculate the area of the smallest circle. The formula for the area of a circle is πr², where r is the radius of the circle. For the smallest circle, the radius is 1 inch. So, the area of the smallest circle is π(1)² = π square inches.
Step 2: Calculate the area of the largest circle. The radius of the largest circle is 3 inches. So, the area of the largest circle is π(3)² = 9π square inches.
Step 3: The probability that a point chosen at random from the largest circle is also in the smallest circle is the ratio of the area of the smallest circle to the area of the largest circle. So, the probability is π/9π = 1/9.
So, the probability that a point chosen at random in the interior of the largest circle is also in the interior of the smallest circle is 1/9.
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