A lighthouse with a rotating beam is located 500m south of a coastal cliff that runs west to east
a) determine a relation for the distance from the lighthouse to the point the light strikes the cliff in terms of the angle of rotation x
b) determine an exact expression for this distance x = 5pi/12
draw a diagram. Clearly
cos(x) = 500/d, so
d = 500/cos(x)
Now just plug in whatever x value you have.
a) To determine a relation for the distance from the lighthouse to the point the light strikes the cliff in terms of the angle of rotation x, we can use trigonometry.
Let's assume that the lighthouse is at the origin (0, 0) on a coordinate system, and the cliff is parallel to the y-axis.
The rotating beam sweeps out a circle with a radius of 500 meters centered at the origin. When the beam is at an angle of x from the positive x-axis, we can think of it as the hypotenuse of a right triangle. The vertical leg of the triangle represents the distance from the lighthouse to the point on the cliff, and the horizontal leg represents the distance from the cliff to the y-axis.
Using trigonometry, we can express the relationship as:
Distance from lighthouse to point on cliff = radius of circle × sine of angle x
Therefore, the relation for the distance from the lighthouse to the point the light strikes the cliff in terms of the angle of rotation x is:
Distance = 500 × sin(x)
b) To determine an exact expression for this distance when x = 5π/12, we substitute this value into the relation we found in part a):
Distance = 500 × sin(5π/12)
Using a calculator, we can evaluate this expression:
Distance ≈ 500 × 0.96592582628
Therefore, the distance from the lighthouse to the point the light strikes the cliff when x = 5π/12 is approximately:
Distance ≈ 482.96 meters
a) To determine the relation for the distance from the lighthouse to the point the light strikes the cliff, we can use trigonometry. Let's consider the angle of rotation x and the distance from the cliff as y.
Since the lighthouse is 500m south of the coastal cliff, we can form a right triangle with the lighthouse, the cliff, and the point where the light strikes the cliff. The vertical leg of this triangle corresponds to the distance y and the hypotenuse corresponds to the 500m distance between the lighthouse and the cliff.
Using trigonometry, we can express the relation between the angle of rotation x and the distance y using the sine function:
sin(x) = y/500
Therefore, the relation for the distance from the lighthouse to the point the light strikes the cliff is y = 500 * sin(x).
b) To find the exact expression for this distance when x = 5π/12, we can substitute the given value into the previous relation:
y = 500 * sin(5π/12)
Using a calculator, we can evaluate this expression:
y ≈ 500 * 0.866 ≈ 433
So, when x = 5π/12, the distance from the lighthouse to the point the light strikes the cliff is approximately 433 meters.