Can someone please explain how to do this problem?
A 450 foot tall building is near a shorter building. A person on top of the shorter building finds the angle of elevation of the roof of the taller building to be 25 degrees and the angle of depression of its base to be 35 degrees. How far apart are the two buildings?
A 450 foot tall building is near a shorter building. A person on top of the shorter building finds the angle of elevation of the roof of the taller building to be 25° and the angle of depression of its base to be 35°. How far apart are the two buildings to the nearest foot? How tall is the shorter building to the nearest foot?
To solve this problem, we need to use trigonometric ratios and the concept of similar triangles. Let's break it down step by step:
Step 1: Visualize the problem
Draw a diagram to understand the situation better. Represent the taller building as a vertical line and the shorter building as a horizontal line. Label the height of the taller building as 450 feet.
Step 2: Identify the relevant angles
The person on top of the shorter building measures two angles:
- The angle of elevation of the taller building's roof, which is 25 degrees.
- The angle of depression of the taller building's base, which is 35 degrees.
Step 3: Formulate a plan
We need to find the distance between the two buildings. To do this, we can focus on the triangle formed by the taller building, the shorter building, and the line of sight of the person on top of the shorter building.
Step 4: Determine the known and unknown quantities
We know:
- The height of the taller building (450 feet),
- The angle of elevation (25 degrees),
- The angle of depression (35 degrees).
We want to find the horizontal distance between the two buildings.
Step 5: Determine the trigonometric functions to use
Since we are dealing with angles and sides of a triangle, we will use trigonometric ratios. Specifically, we will use:
- Tangent (tan) for the angle of elevation, as the roof's height (450 feet) is on the opposite side and the horizontal distance between the buildings is on the adjacent side.
- Tangent (tan) for the angle of depression, as the base's height (450 feet) is on the opposite side and the horizontal distance between the buildings is on the adjacent side.
Step 6: Apply trigonometry
Let's start with the angle of elevation:
tan(angle of elevation) = opposite / adjacent
Using the known values:
tan(25 degrees) = 450 feet / adjacent
Now, let's solve for the adjacent side:
adjacent = 450 feet / tan(25 degrees)
Using a calculator, we find that the adjacent side is approximately 1066.48 feet.
Next, let's move on to the angle of depression:
tan(angle of depression) = opposite / adjacent
Using the known values:
tan(35 degrees) = 450 feet / adjacent
Solving for the adjacent side:
adjacent = 450 feet / tan(35 degrees)
Using a calculator, we find that the adjacent side is approximately 568.19 feet.
Step 7: Determine the distance between the buildings
To find the horizontal distance between the buildings, we need to subtract the two adjacent sides:
Distance = 1066.48 feet - 568.19 feet
Therefore, the distance between the two buildings is approximately 498.29 feet.
So, the two buildings are approximately 498.29 feet apart.