Respuesta :
the angle be measured for the percentage error in estimating the height of the building to be less than 2 percent is 5.8.
In all the computations below, θ will be the angle of elevation, measured in radians.
The if the answer is desired in radians, we can convert it at the end. We want to use
radians so that the derivatives of trigonometry functions are the ones we have memorized. Let
h denote the height of the building, measured in feet. Then
tan θ = h/10, so that
h = 10 tan θ and dh = 10 sec²θdθ.
It follows that the percentage error is approximately
dh/h =30 sec²θdθ/ 30 tan θ
= 1cos² θ /sin θcosθ
=dθsin θ cos θ
We want this percentage error less than 2%, so dθ/sin θ cos θ < 0.04.
It follows that (using double-angle formula) we want
dθ < 0.04 sin θ cos θ = 0.02 sin(2θ).
Since 150◦ = 150( π180 )
= 5π/6 radians,
we want dθ < 0.02 sin5π/6
= 0.02( 1/2)
= 0.01.
In other words, we need to measure the angle to within 0.01 radians, or
0.01 (180/π)
≈ 0.572957795130823208767981548141◦
rounding up 0.58.
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