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.
To find the total area covered by the two circles, we need to calculate the areas of each circle separately and then add them together.
Let's start with the smaller circle:
The radius of the smaller circle is half of the diameter, which is equal to half the distance between points A and B. Since they are 4 inches apart, the radius of the smaller circle is 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. So, for the smaller circle:
A1 = π(2^2)
= π(4)
= 4π square inches
Now let's move on to the larger circle:
The larger circle overlaps the smaller circle and has a quarter-circle arc from A to B along its circumference. This means the diameter of the larger circle is equal to the distance between points A and B plus the radius of the smaller circle.
The diameter of the larger circle is equal to 4 inches (distance between A and B) + 2 inches (radius of the smaller circle), which is 6 inches.
The radius of the larger circle is half of its diameter, which is 6 inches divided by 2, giving us 3 inches.
The area of the larger circle is therefore:
A2 = π(3^2)
= π(9)
= 9π square inches
To find the total area covered by the two circles, we add the individual areas together:
Total area = A1 + A2
= 4π + 9π
= 13π square inches
Thus, the total area covered by the two circles is 13π square inches.
To find the total area covered by the two circles, we need to calculate the individual areas of each circle and then add them together.
First, let's find the area of the smaller circle. Since segment AB is the diameter of this circle, we can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.
Since the diameter of the smaller circle is 4 inches, the radius is half of that, which is 2 inches. Plugging this value into the formula, we have:
A1 = π(2 inches)^2
A1 = 4π square inches
Now, let's find the area of the larger circle. The arc from A to B along its circumference is a quarter-circle, so we can calculate the area of this quarter-circle and subtract it from the area of the whole circle to find the remaining covered area.
The length of the arc AB in the larger circle is equal to the circumference of the whole circle since it covers the entire circle's circumference. The circumference formula is C = 2πr, where C is the circumference and r is the radius.
Since the diameter of the larger circle is 4 inches, its radius is 2 inches. Plugging this value into the circumference formula, we have:
C2 = 2π(2 inches)
C2 = 4π inches
The length of the arc AB is equal to one-fourth of the total circumference of the larger circle, which means it is 1/4 * 4π = π inches.
To find the area of the quarter-circle, we can use the formula A = (1/4)πr^2, where A is the area and r is the radius.
Since the radius of the larger circle is 2 inches, plugging this value into the formula, we have:
A2 = (1/4)π(2 inches)^2
A2 = (1/4)4π square inches
A2 = π square inches
The total area covered by the two circles is the sum of the area of the smaller circle (A1 = 4π square inches) and the area covered by the larger circle excluding the quarter-circle (A2 = π square inches). Adding these two areas together, we have:
Total area = A1 + A2
Total area = 4π + π
Total area = 5π square inches
Therefore, the total area covered by the two circles is 5π square inches.