how do I set up this problem a man standing on the deck of a ship with his eyes 25 feet above sea level notes that the angle of elevation to the top of a 25 foot lighthouse is 3 degrees 27 minutes. how far is the boat from the lighthouse
Strange wording.
How high is the lighthouse above the sea level ?
you can see why Reiny's asking his question. If the base of the lighthouse is at sea level, then the man's eyes are at the same height as the top of the lighthouse, and the angle of elevation will be zero.
So, if the base of the lighthouse is at height h, then the top of the lighthouse is h above the man's eyes. So, the distance d of the boat can be found by
h/d = tan 3°27'
looks like something's amiss with the wording of the problem.
Two boats traveling toward a lighthouse that is 20o ft above sea level at its top. When the 2 boats and the lighthouse are collinear, the boats are exactly 250 ft apart and the boat closest to the lighthouse of 15° as shown.
Find the measure of x, rounded to the nearest hundredth.
To solve this problem, you can use trigonometry. Let's break it down into steps:
Step 1: Draw a diagram
Draw a diagram to visualize the situation. Draw a horizontal line to represent the sea level, and another vertical line to represent the height of the lighthouse. Mark the point where the man is standing on the ship, the lighthouse, and the angle of elevation.
Step 2: Identify known and unknown values
From the given information, we know that the height of the man's eyes above sea level is 25 feet, and the angle of elevation to the top of the lighthouse is 3 degrees 27 minutes. The unknown value is the distance between the boat and the lighthouse.
Step 3: Use trigonometry
In this case, we can use the tangent function, which relates the angle of elevation to the height and distance. The formula is:
tangent(angle) = opposite/adjacent
In this case, the angle is 3 degrees 27 minutes, and the opposite side is the height of the lighthouse (25 feet). The adjacent side is the distance from the man to the lighthouse (which we want to find).
Step 4: Convert the angle to decimal form
To use the tangent function, we need to convert the angle of 3 degrees 27 minutes to decimal form. There are 60 minutes in a degree, so the conversion is:
3 degrees + (27 minutes / 60) = 3.45 degrees
Step 5: Solve for the distance
Using the tangent function, we can set up the equation:
tangent(3.45 degrees) = 25 feet / distance
Now, solve for the distance by isolating it in the equation:
distance = 25 feet / tangent(3.45 degrees)
Step 6: Calculate the distance
Using a calculator or trigonometric table, find the tangent of 3.45 degrees and calculate the distance:
distance ≈ 25 feet / 0.0603
distance ≈ 414.71 feet
Therefore, the boat is approximately 414.71 feet away from the lighthouse.