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.? AB is not the diameter of the larger circle

Answer 1

First, let's draw a diagram depicting the situation described in the problem:

```
___________________________________________
| * * * | *
| * * | *
| * * | *
| * * | * B
| * . . . . . . . . . . . . * | *
| * . | *
| * * * * * * * * * . | *
| * * * * | *
| * * * * * * * | *
| * | | A *
| * | | *
| * | | *
| * | | *
| * | *
| *__________|_____________*
```

In the diagram above, point A and point B are marked as two blue dots, representing the endpoints of segment AB. The first circle is drawn with AB as its diameter, and the second circle is drawn as a larger circle, overlapping the first circle.

To find the total area covered by the two circles, we need to calculate the sum of the individual areas of both circles.

The area of a circle can be calculated using the formula: A = πr^2, where A represents the area and r represents the radius of the circle.

In the first circle, the radius (r1) is equal to half the length of AB, which is 4 inches divided by 2, resulting in r1 = 2 inches.

The area of the first circle is A1 = π(2 inches)^2 = 4π square inches.

In the second circle, the radius (r2) can be determined by finding the distance from the center of the circle to point A or B. Since the quarter-circle arc AB is drawn along the circumference, this means that the center of the second circle lies exactly at the midpoint of AB.

Since AB is 4 inches long, the midpoint is located 2 inches from point A (or point B). Thus, the radius (r2) of the second circle is also 2 inches.

The area of the second circle is A2 = π(2 inches)^2 = 4π square inches.

The total area covered by the two circles is the sum of their individual areas:

Total area = A1 + A2 = 4π + 4π = 8π square inches.

Therefore, the total area covered by the two circles is 8π square inches.