Two boats are spotted by a Coast Guard helicopter. The helicopter is 150 yards above the water. The angles of depression are 12.5° and 15.3°. How far apart are the boats? (The boats are in the same general direction from the helicopter.) Pls help i have 7 of these types of questions and i have gotten them all wrong..
First is always draw a diagram. Then review your basic trig functions. It should be clear that the distance between the boats is
150 cot 12.5° - 150 15.3°
You will also see this problem where you want to find the height, given the distance between two points on the ground.
275.46
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Let's call the distance between the helicopter and the closer boat "x" and the distance between the helicopter and the farther boat "y". We know that the angles of depression to the closer and farther boat are 12.5° and 15.3° respectively.
Using the tangent function, we can set up two equations:
tan(12.5°) = x / 150 --> Equation 1
tan(15.3°) = y / 150 --> Equation 2
Now we can solve these equations to find the values of x and y.
Step 1: Solve Equation 1 for x:
Multiply both sides of Equation 1 by 150:
x = 150 * tan(12.5°)
Step 2: Solve Equation 2 for y:
Multiply both sides of Equation 2 by 150:
y = 150 * tan(15.3°)
Step 3: Calculate the values of x and y using a calculator:
x ≈ 32.94 yards (rounded to 2 decimal places)
y ≈ 39.65 yards (rounded to 2 decimal places)
Step 4: Calculate the distance between the two boats:
The distance between the two boats can be found by subtracting the distances from the helicopter to each boat.
Distance = y - x
Distance ≈ 39.65 - 32.94
Distance ≈ 6.71 yards (rounded to 2 decimal places)
Therefore, the boats are approximately 6.71 yards apart.