# Computing error limits with total differential.

## Problem

Tower height is measured with angle measurement from two points $A$ and $B$ which are in same same direction (relative to the tower). Measured angles are $50\pm1$ degree,$35\pm 1$ degree and length between points $A$ and $B$ is measured $100\pm1$ meters. What is tower height and how large error there can be in tower height with these measurements ?

## Attempt to solve

Since both of these are on the same direction we can think of this as two dimensional situation. I call the further point $A$ an the closer one $B$. Also corresponding angle to these points are angles $\alpha$ for point $A$ and $\beta$ for point $B$. I also refer the length between point $A$ and $B$ as $d$ and the length between point $B$ and the tower as $w$. Tower height is $h$

A equation for tower length can be derived from trigonometric functions. $$\begin{cases}\tan({\alpha})=\frac{h}{d+w} \\ \tan({\beta})=\frac{h}{w} \end{cases} \\$$

we can now solve $h$ from these.

$$w=\frac{h}{\tan({\beta})}$$ $$\tan(\alpha)=\frac{h}{d+\frac{h}{tan(\beta)}}$$ $$h=\frac{-d\sin(\alpha)\sin(\beta)}{\sin(\alpha-\beta)}$$

Now by simply pluggin in values we should get approximate for tower length.

$$h\approx\frac{-100\cdot \sin(35)\cdot \sin(50)}{\sin(35-50)}\approx 169.7653 \text{ m}$$ Now since this is only approximate we want to know error margins for height of the tower. One way to evaluate this is to compute total differential. $$\Delta h = |\frac{\delta h}{\delta \alpha}|\Delta \alpha+|\frac{\delta h}{\delta \beta}|\Delta \beta + |\frac{\delta h}{\delta d}|\Delta d$$ Computing partial derivatives of $h$ with respect to all the variables that have error margins.

$$\Delta h = |\frac{d\sin({\alpha})\sin({\beta})}{\tan(\alpha-\beta)\sin(\alpha-\beta)}-\frac{d\cos(\alpha)\sin(\beta)}{\sin(\alpha-\beta)}|\Delta \alpha + |-\frac{d\sin(\alpha)\cos(\beta)}{\sin(\alpha-\beta)}-\frac{d\sin(\alpha)\sin(\beta)}{\tan(\alpha-\beta)sin(\alpha-\beta)}|\Delta \beta+ |-\frac{\sin(\alpha)\sin(\beta)}{\sin(\alpha-\beta)}|$$

now if i try to plug in values

$$\Delta \alpha=1,\Delta \beta =1, \Delta d = 1$$ (These are error margins that are given in the problem description.) and these $$d=100,\alpha=35,\beta=50$$ i get $$\Delta h \approx 591.6512278 \text{ m}$$ which clearly cant be true. Now if someone cant notice the flaw in this that would be highly appreciated.

• Always use the angles converted in radians... The derivative, when the angles are in degrees are not like the ones you wrote. – N74 Jan 29 '18 at 20:08