Final answer:
To find the height of an office building using the angles of elevation and depression, trigonometry is used. The height is calculated by adding the distance above and below the observer's eye line, which are derived using the tangent of the given angles and the distance from the person to the building.
Step-by-step explanation:
To determine the height of the office building using the angle of elevation and the angle of depression, we can use trigonometric functions. The angle of elevation to the top of the building is 23 degrees, and the angle of depression to the base is 50 degrees, with the person observing from 500 feet away.
Let's denote the height of the building as H, the distance to the building as D (500 ft), the angle of elevation as α (23 degrees), and the angle to the base as β (50 degrees). Using the tangent function, which relates angles to the ratio of opposite over adjacent sides in a right triangle, we can write the following equations:
tan(α) = H / D
tan(β) = base_height / D
Thus, H = D * tan(α) and base_height = D * tan(β). We can now find the height of the building by adding the height above the observer's eye level (H) to the height from the base of the building to the observer's eye level (base_height).
Step 1: Calculate H = 500 ft * tan(23°) = 500 ft * 0.4245 (using a calculator)
Step 2: Calculate base_height = 500 ft * tan(50°) = 500 ft * 1.1918 (using a calculator)
Step 3: Add both heights to find total height of the building: Total height = H + base_height
By carrying out the calculations, you will find the exact height of the office building, which is the answer we seek.