The angle of depression from the top of a 75 ft. lighthouse to a ship out in the ocean is 40*. How far out is the ship from the lighthouse? I've been stuck on this problem for an hour, I know the answer but don't know how to set it up/enter it in a calculator.
And therein lies the problem, the book says it's 6445.5 ft. or 1.2 miles... Oh dear, it's 40' not 40*
Got it. Thanks Reiny. It's 75/sin.667 .40x60=.667
And that comes out to the 6445.5 I was looking for
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In this case, the angle of depression is 40 degrees, which is the angle between the horizontal line from the top of the lighthouse to the ship and the downward line from the top of the lighthouse to the ship. The lighthouse is the reference point.
Given that the height of the lighthouse is 75 ft, we can consider the vertical line from the top of the lighthouse to the ship as the opposite side, and the distance from the lighthouse to the ship as the adjacent side.
Now, we need to find the length of the opposite side (the vertical line). We can use the trigonometric identity:
tan(angle) = opposite / adjacent
tan(40 degrees) = 75 / adjacent
To find the length of the adjacent side (the distance from the lighthouse to the ship):
adjacent = 75 / tan(40 degrees)
Now, we can use a scientific calculator to evaluate this expression. Enter the values as follows:
75 / tan(40)
After evaluating this expression, the result will give you the distance from the lighthouse to the ship.
Note: Make sure your calculator is set to degrees mode before performing this calculation.
I hope this explanation helps you to set up and solve the problem! Let me know if you have any further questions.
An hour??
Did you make a sketch?
Mine is a right-angled triangle with base angle of 40°, a horizontal of x, and a vertical of 75 ft, and a hypotenuse of h
"How far out is the ship from the lighthouse" is a vague question.
Do you want the direct distance between the lighthouse and the ship?
then sin40° = 75/h
h = 75/sin40° = appr 116.68
If you want the horizontal distance between the boat and the lighthouse, ...
tan40° = 75/h
h = 75/tan40 = appr 89.38