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.?
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:
```
_____/_____________\_____
/ \
/ \
/ \
/ \
/ \
\ /
\ /
\ /
\ /
\_______________/
```
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.
To find the total area covered by the two circles, we need to find the area of each circle and then add them together.
Let's start by finding the area of the smaller circle. The diameter of the smaller circle is the length of segment AB, which is 4 inches. The radius (r) of the smaller circle is half of the diameter, so r = 4/2 = 2 inches.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. Plugging in the value of r, we have A1 = π(2)^2 = 4π square inches.
Next, let's find the area of the larger circle. The arc from A to B along its circumference is a quarter-circle. This means that the length of the arc is one-fourth of the circumference of the larger circle.
The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. In this case, the circumference of the larger circle is 4 inches, as given in the problem. So, 2πr = 4 inches.
Solving for r, we have r = 4/(2π) = 2/π inches.
Since the arc forms a quarter-circle, the angle at the center of the circle is 90 degrees. This means that the area of the shaded region is 1/4 of the area of the full circle.
The area of the larger circle is given by the formula A = πr^2. Plugging in the value of r, we have A2 = π(2/π)^2 = π/π^2 = 1/π square inches.
Now, let's add the areas of the two circles: A1 + A2 = 4π + 1/π = (4π^2 + 1)/π square inches.
Therefore, the total area covered by the two circles is (4π^2 + 1)/π square inches.
Here is a visual representation of the two circles:
```
_________
/ \
/ \
/ A B\
| o-------o |
| | | |
| | | |
\ | | /
\ | |/
\_________\
```
To find the total area covered by the two circles, let's determine the areas of each circle individually and then add them up.
1. The smaller circle:
Since the segment AB is a diameter of the smaller circle, its radius is half the length of AB, which is 4/2 = 2 inches.
The formula to calculate the area of a circle is given as A = πr^2, where A represents the area and r represents the radius.
Therefore, the area of the smaller circle is A1 = π(2^2) = 4π square inches.
2. The larger circle:
The arc from A to B along the circumference forms a quarter-circle. The diameter of this quarter-circle is also the same as AB, which is 4 inches.
To find the area of the quarter-circle, we need to calculate the circumference first. The formula for the circumference of a circle is given as C = 2πr, where C represents the circumference and r represents the radius.
Since the diameter is 4 inches, the radius of the larger circle is 4/2 = 2 inches.
Therefore, the circumference of the larger circle is C2 = 2π(2) = 4π inches.
As the arc forms a quarter-circle, its length is one quarter of the full circumference, which is 1/4 * (4π) = π inches.
To calculate the area of the quarter-circle, we use the formula A = (πr^2)/4, where A represents the area and r represents the radius.
So, the area of the quarter-circle is A2 = (π(2^2))/4 = (4π)/4 = π square inches.
Now, to find the total area covered by the two circles, we add the areas of the smaller and larger circles together:
Total area = A1 + A2 = 4π + π = 5π square inches.
As for the visual representation, unfortunately, as a text-based AI, I am unable to draw images or provide visual content. However, you can easily visualize the scenario by drawing two circles with one being smaller and having a diameter of 4 inches while the other circle overlaps it with an arc from A to B forming a quarter-circle.