The equation of the circle, x² + y² = 5625, describes the broadcast area of a radio station, indicating that the station reaches listeners within a circular region. To interpret this equation clearly, we can relate it to the standard form of a circle’s equation, which is (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and r is the radius.
In this case, since the equation does not have h or k terms, we can infer that the circle is centered at the origin, or point (0, 0), of the coordinate plane. The constant 5625 represents the square of the radius. To find the radius (r), we take the square root of 5625:
r = √5625 = 75
This means the radio station has a broadcast radius of 75 miles. Consequently, the station covers a circular area where every point within this radius can receive its signal.
The area of the broadcast region can also be calculated using the formula for the area of a circle, which is A = πr². Plugging in our radius, we find:
A = π(75)² = π(5625) ≈ 17671.46 square miles
This substantial area highlights the potential reach of the radio station, allowing it to connect with a large audience across various locations, thereby maximizing its listener base and advertisements. With a clear understanding of the broadcast area, the radio station can strategize its programming and marketing efforts more effectively to engage listeners within this expansive region.