The rotating spotlight from the Coast Guard ship can illuminate up to a distance of 250 m. An observer on the shore is 500 m from the ship. HIs line of sight to the ship makes an angle of 20 degrees with the shoreline. What length of shoreline is illuminated by the spotlight?
(This is how far I have gotten)
sinA = sinB
----- -----
a b
sinA sin20
---- = -----
500 250
sinA = 500(sin20)
---------
250
sinA = 43 degrees
the ship is 500 sin 20° = 171 m from shore.
A beam of 250m will illuminate 2√(250^2-171^2) = 365 m of shoreline.
Thank you
To find the length of the shoreline illuminated by the spotlight, we can use the trigonometric relationships in a right triangle.
Let's label the length of the shoreline illuminated by the spotlight as "x."
We know that the angle between the observer's line of sight and the shoreline is 20 degrees. This angle is opposite to the side of length x.
Using the sine function, we can set up the following equation:
sin(20 degrees) = x / 500 m
Rearranging the equation to solve for x, we have:
x = 500 m * sin(20 degrees)
Calculating this value, we find:
x ≈ 171.01 m
Therefore, the length of the shoreline illuminated by the spotlight is approximately 171.01 meters.
To calculate the length of the shoreline that is illuminated by the spotlight, you can use trigonometry and the concept of similar triangles.
First, let's define the variables:
- Let x be the length of the shoreline that is illuminated.
- Let y be the distance from the observer to the illuminated part of the shoreline.
Now, let's set up a proportion using similar triangles:
(sin A) / y = (sin B) / x
Here, A is the angle between the observer's line of sight and the shoreline, and B is the angle between the spotlight's line of sight and the shoreline. The given information tells us that A is 20 degrees.
Plugging in the values we know:
(sin 20) / y = (sin 90) / 250
Since sin 90 is equal to 1, the equation simplifies to:
(sin 20) / y = 1 / 250
Now, let's isolate y by cross-multiplying:
y = 250 / sin 20
Using a calculator, you can evaluate sin 20 to find its decimal value.
Next, plug that decimal value into the equation to calculate y.
Once you have found the value of y, you can subtract it from 500 (the total distance from the observer to the ship) to find the length of the shoreline that is illuminated.
I hope this explanation helps you understand how to approach this problem!