A person is standing 90 feet away from the Space Needle. The person looks up at the Space Needle at an angle of elevation of 81 degrees. The distance from the floor to the person's eye level is 5 feet. What must be the height of the Space Needle?
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To find the height of the Space Needle, we can use trigonometry and the concept of similar triangles. Let's break down the problem step by step:
Step 1: Visualize the scenario.
Imagine a right-angled triangle formed by the person, the Space Needle, and a line connecting them. The line connecting the person's eye level and the top of the Space Needle represents the height we want to find.
Step 2: Identify the given information.
We know the following:
- The person is standing 90 feet away from the Space Needle.
- The person's angle of elevation is 81 degrees.
- The distance from the floor to the person's eye level is 5 feet.
Step 3: Label the diagram.
Label the triangle based on the given information. Let's call the height of the Space Needle "h" (which we want to find) and the distance from the person to the Space Needle's base "90 feet."
Now, let's focus on the angle of elevation. In a right-angled triangle, the angle of elevation is the angle between the horizontal ground and the line of sight to an object above the ground. In this case, it is 81 degrees between the horizontal line (the ground) and the line of sight from the person to the top of the Space Needle.
Step 4: Set up the trigonometric relationship.
Here, we are dealing with the tangent function, as tangent represents the ratio of the opposite side and the adjacent side of a right-angled triangle. In this case, the opposite side is the height "h," and the adjacent side is the distance from the person to the Space Needle's base, which is 90 feet.
Thus, we have the equation: tan(81 degrees) = h / 90 feet.
Step 5: Solve for the unknown, "h."
To solve the equation, we rearrange it to isolate "h." Multiply both sides of the equation by 90 to get: h = 90 * tan(81 degrees).
Using a calculator, evaluate tan(81 degrees), which is approximately 7.11537. Multiply that by 90 feet to get the height, h = 7.11537 * 90 feet.
The height of the Space Needle is approximately 639.38 feet.
Therefore, the height of the Space Needle must be around 639.38 feet.