A radio station has a broadcast area in the shape of a circle with equation 22 + y?= 5,625, where the constant represents square miles.
a. Find the intercepts of the graph.
b. State the radius in miles.
c. What is the area of the region in which the broadcast from the station can be picked up?
a. To find the intercepts of the graph, we set y = 0 to find the x-intercepts and x = 0 to find the y-intercepts:
1. x-intercept:
22 + 0 = 5,625
x = 5,625
So, the x-intercept is (5,625, 0).
2. y-intercept:
22 + y = 5,625
y = 5,625 - 22
y = 5,603
So, the y-intercept is (0, 5,603).
b. To find the radius of the circle, we need to solve for y in terms of x:
22 + y = 5,625
y = 5,625 - 22
y = 5,603
The radius is the distance from the center of the circle to any point on the circle. In this case, the center of the circle is at the point (0, 5,603), and the y-coordinate is the radius. Therefore, the radius is 5,603 miles.
c. The area of a circle is given by the formula A = πr^2, where r is the radius. In this case, the radius is 5,603 miles. So, the area of the region in which the broadcast from the station can be picked up is:
A = π(5,603)^2
A = 98,862,339π
A ≈ 310,982.57 square miles
Therefore, the area of the region in which the broadcast from the station can be picked up is approximately 310,982.57 square miles.