A surveyor, standing 80 ft. from the base of a building, measures the angle of elevation to the top of the building to be 75 degrees. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%?
To determine how accurately the angle must be measured, we need to consider the percentage error in estimating the height of the building.
Let's start by calculating the actual height of the building. We can use trigonometry.
We have a right-angled triangle formed by the surveyor's line of sight, the ground, and the height of the building. The angle of elevation is 75 degrees, and the distance from the surveyor to the base of the building is 80 ft.
Using the definition of tangent, we can set up the following equation:
tan(75 degrees) = height of the building / distance to the base
tan(75 degrees) = h / 80
To find the height of the building, we can rearrange the equation:
h = 80 * tan(75 degrees)
Calculating this using a calculator, we find:
h ≈ 292.012 ft
Now let's consider the allowed percentage error, which is less than 4%. To calculate the maximum error in the height estimation, we multiply the actual height of the building by 4%:
4% = 0.04
allowed_error = 0.04 * actual height
allowed_error = 0.04 * 292.012
allowed_error ≈ 11.68048 ft
Therefore, the maximum allowed error in estimating the height of the building is approximately 11.68048 ft.
To find out how accurately the angle must be measured, we need to determine the maximum allowable change in the angle that gives us an error of 11.68048 ft.
The key observation here is that the error (11.68048 ft) is directly related to the change in the angle of elevation. As the angle changes, the height estimation changes.
Let's assume the angle changes by a small amount dθ. Then, the corresponding change in the height estimation is given by:
Δh = 80 * tan(75 degrees + dθ) - 80 * tan(75 degrees)
To simplify the calculation, we can use the small angle approximation:
tan(75 degrees + dθ) ≈ tan(75 degrees) + dθ * sec^2(75 degrees)
Using this approximation, we can rewrite the equation for the change in the height estimation:
Δh ≈ 80 * (tan(75 degrees) + dθ * sec^2(75 degrees)) - 80 * tan(75 degrees)
Simplifying further, we get:
Δh ≈ 80 * dθ * sec^2(75 degrees)
We want the error in height estimation (Δh) to be within the allowed error (11.68048 ft). Therefore, we can set up the following inequality:
80 * dθ * sec^2(75 degrees) ≤ allowed_error
Substituting the values, we get:
80 * dθ * sec^2(75 degrees) ≤ 11.68048 ft
To find the maximum allowable change in the angle of elevation (dθ), we need to solve for it:
dθ ≤ 11.68048 ft / (80 * sec^2(75 degrees))
Calculating this using a calculator, we find:
dθ ≤ 0.00325507 degrees
Therefore, the angle of elevation must be measured with an accuracy of at least 0.00325507 degrees (or better) to ensure the percentage error in estimating the height of the building is less than 4%.