Amy marks two points A and B that are 4 inches apart. She draws one circle that has segment AB as a diameter. She then draws a larger circle, which overlaps the first circle, such that the arc from A to B along its circumference is a quarter-circle. What is the total area covered by the two circles? Express your answer in terms of pi. Could you draw a visual picture of this.?

Answer 1

To find the total area covered by the two circles, we first find the area of each individual circle.
The first circle has a diameter of 4 inches, so its radius is half of the diameter, which is 4/2 = 2 inches. The area of this circle is given by the formula A = πr^2, where A is the area and r is the radius. Substituting the value of the radius, we have A1 = π(2^2) = 4π square inches.

The second circle has a diameter equal to the arc AB, which has a length of 4 inches. Since the arc is a quarter-circle, the diameter is the same as the radius of a full circle. Hence, the radius of the second circle is also 4 inches. The area of this circle is A2 = π(4^2) = 16π square inches.

The total area covered by the two circles is the sum of their individual areas. So, the total area is A1 + A2 = 4π + 16π = 20π square inches.

Here is a visual representation of the two circles:

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The small circle corresponds to the first circle, and the larger circle corresponds to the second circle. The two circles overlap such that AB is the diameter of both circles.