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%?
gaon this problem can't be solved as it is a naughty question!
To determine how accurately the angle must be measured for the percentage error in estimating the height of the building to be less than 4%, we can use trigonometry.
First, let's define the variables:
Let h be the height of the building.
Let d be the distance from the surveyor to the base of the building (80 ft).
Let θ be the angle of elevation to the top of the building (75 degrees).
Now, we can set up the trigonometric relationship between these variables. We know that:
tan(θ) = h/d
Rearranging the equation, we have:
h = d * tan(θ)
Next, we need to calculate the maximum error in estimating the height of the building. We are given that the percentage error should be less than 4%, which can be written as 0.04 in decimal form. Therefore, the maximum error (e) can be calculated as:
e = 0.04 * h
Substituting h = d * tan(θ), we get:
e = 0.04 * d * tan(θ)
Now, we can solve for the angle θ. Rearranging the equation, we have:
tan(θ) = e / (0.04 * d)
Taking the inverse tangent of both sides, we find:
θ = arctan(e / (0.04 * d))
Substituting the given values, we have:
θ = arctan(e / (0.04 * 80))
Finally, we can calculate the value of θ, which represents the maximum angle of elevation that can be measured for the percentage error in estimating the height of the building to be less than 4%.
Please note that the actual percentage error will depend on the measured angle of elevation and might be smaller than the maximum angle calculated here.