Question

A right triangle has base b = 100 plusminus 1 ft and adjacent angle theta = 30 degree plusminus 0.5 degree. Calculate the height h and its uncertainty.

Answer 1

Answer:

The height h of this right triangle is 57.74±1.74ft.

Explanation:

Knowing that this is a right triangle, if one of the angles adjacent to the base, θ is 30º, the other angle is 90º. Therefore we can calculate the height h can be calculated with the tangent:

$tan(\theta)=\frac{h}{b}\leftrightarrow h=tan(\theta)b=tan(30^\circ)100ft=57.74ft$

For the uncertainty we use the  partial derivatives:

$\Delta h=|\frac{dh}{db}|\Delta b+ |\frac{dh}{d\theta}|\Delta \theta\\\Delta h=|tan(\theta)|\Delta b+ |\frac{b}{cos^2(\theta)}|\Delta \theta$

We have to be careful to use Δθ in radians:

$\Delta h=|tan(30^\circ)|1ft+ |\frac{100ft}{cos^2(30^\circ)}|\frac{0.5^\circ2\pi}{360^\circ}=1.74ft$