Final answer:
To estimate the distance to the radio tower using the arc length formula, we convert the 3-degree angle to radians and solve for the radius (distance) given the tower's height as the arc length. The closest estimate for the distance is approximately 1052 feet.
Step-by-step explanation:
The question asks us to estimate the distance between the observer and a 55-foot tall radio tower, given the angle it subtends from the observer is 3 degrees. To solve this, we use the arc length formula, which in a circle relates the arc length (s), the radius (r), and the subtended angle in radians (θ) through the equation s = rθ.
First, we need to convert the angle from degrees to radians. We know that 180 degrees is equivalent to π radians. Therefore, 3 degrees is (3/180)π radians.
Since the height of the tower (which is equivalent to the arc length in this scenario) is 55 feet, the formula becomes 55 = r(3/180)π. To find the distance r, we divide both sides of the equation by (3/180)π, resulting in r ≈ 1052 feet.
While 1052 is not one of the provided options, the closest answer by rounding to the nearest foot is approximately 1050 feet. Since this option is not listed, it appears that there might be an error in the given answer choices, or additional information might be required to select one of the provided options accurately.