Daniel sees a lighthouse in the harbor. He estimates the angle of elevation is 70°. If the lighthouse is 120 feet tall, what is the approximate distance between Daniel and the top of the lighthouse? (Assume the lighthouse meets the ground at a right angle.)
To find the approximate distance between Daniel and the top of the lighthouse, we can use trigonometry. In this case, we need to use the tangent function because we have an angle and a side length of a right triangle.
Step 1: Recall that the tangent function is defined as the opposite side divided by the adjacent side in a right triangle.
tan(angle) = opposite side / adjacent side
Step 2: Let's assign variables to the lengths we know. Let x be the distance between Daniel and the top of the lighthouse. The height of the lighthouse is given as 120 feet.
Step 3: Identify the sides of the right triangle. The height of the lighthouse is the opposite side, and the distance between Daniel and the lighthouse is the adjacent side.
Step 4: Write the equation using the tangent function:
tan(70°) = 120 feet / x
Step 5: Solve for x by isolating it on one side of the equation.
x = 120 feet / tan(70°)
Step 6: Calculate the value of x using a calculator.
x ≈ 120 feet / 2.7474
Therefore, the approximate distance between Daniel and the top of the lighthouse is approximately 43.72 feet.
To find the approximate distance between Daniel and the top of the lighthouse, we can use basic trigonometry. We can use the tangent function, which relates the angle of elevation to the distance and height.
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 tangent of the angle of elevation (70 degrees) is equal to the height of the lighthouse (120 feet) divided by the distance between Daniel and the lighthouse.
Let's call the distance between Daniel and the lighthouse "d."
So, we have the equation: tan(70°) = 120 feet / d
To solve for "d," we can rearrange the equation as follows:
d = 120 feet / tan(70°)
Now, let's calculate the approximate distance using this formula:
d ≈ 120 feet / tan(70°)
Using a calculator:
d ≈ 120 feet / 2.7474774195
d ≈ 43.675 feet
Therefore, the approximate distance between Daniel and the top of the lighthouse is approximately 43.675 feet.