Since its construction in 1821, Marblehead Lighthouse has stood at
the entrance to Sandusky Bay, guiding sailors along Lake Erie’s rocky
shores. The 65-foot tower is one of Ohio’s best-known landmarks and
the oldest continuously operating lighthouse on the Great Lakes.
1. The range of a lighthouse is the maximum distance at which its
light is visible. In the figure, point A is the farthest point from
which it is possible to see the light at the top of the lighthouse L.
The distance along Earth s is the range. Assuming that the radius
of Earth is 4000 miles, find the range of Marblehead Lighthouse.
To find the range of Marblehead Lighthouse, we need to use the concept of the Earth's curvature. The range of a lighthouse is determined by the line of sight between the viewer and the top of the lighthouse.
In this case, point A represents the farthest point from which it is possible to see the light at the top of the lighthouse. We can consider point A as being at the horizon.
To calculate the range, we can use the formula for the distance along the surface of a sphere, which is given by:
Distance = radius of the sphere * angle in radians
Since the radius of Earth is given as 4000 miles, we can use this value in the formula.
The angle in radians can be found using basic trigonometry. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (65 feet) and the adjacent side is the radius of Earth (4000 miles).
Using the equation tan(theta) = opposite/adjacent, we can solve for theta:
tan(theta) = 65 / 4000
Taking the inverse tangent of both sides, we get:
theta = arctan(65 / 4000)
Now, we can substitute the value of theta into our distance formula:
Distance = 4000 * arctan(65 / 4000)
Calculating this expression will give us the range of Marblehead Lighthouse in miles.