Distance between prisoners? From the top of a tower $75$ m high, a guard sees two prisoners, both due west of him. If the angles of depression of the two prisoners are $10^{\circ}$ and $17^{\circ}$, calculate the distance between them.
I did $(\tan(17^{\circ})=\frac{75}{x})-(\tan(10^{\circ})=\frac{75}{y})=180$ but someone told me that the answer is $9.7$.
Which one of us is right?
 A: Your answer is correct, but your solution is not. 
Consider the diagram below:

Since alternate interior angles are congruent, the angle of elevation from the prisoner to the guard is congruent to the angle of depression from the guard to the prisoner.  
Observe that 
\begin{align*}
\tan(17^\circ) & = \frac{75~\text{m}}{x} \tag{1}\\
\tan(10^\circ) & = \frac{75~\text{m}}{y} \tag{2}
\end{align*}
where $x$ is the distance from the base of the guard tower to the nearer of the two prisoners and $y$ is the distance from the base of the guard tower to the farther of the two prisoners.  We need to solve for solve the distance between the prisoners, which is $d = y - x$.  Solving equation 1 for $x$ yields
$$x = \frac{75~\text{m}}{\tan(17^\circ)}$$
Solving equation 2 for $y$ yields
$$y = \frac{75~\text{m}}{\tan(10^\circ)}$$
Hence, the distance between the prisoners is 
$$d = y - x = \frac{75~\text{m}}{\tan(10^\circ)} - \frac{75~\text{m}}{\tan(17^\circ)} \approx 180~\text{m}$$
The person who told you that the answer was $9.7~\text{m}$ incorrectly obtained 
$$d = 75~\text{m}[\tan(17^\circ) - \tan(10^\circ)]$$  
In writing your solution, you incorrectly wrote that 
$$d = \tan(17^\circ) - \tan(10^\circ) = \frac{75~\text{m}}{x} - \frac{75~\text{m}}{y}$$
which is not what you meant.
A: HINT
You are right in considering difference between x and y. But why did tower height decrease to 70?please correct it.
$$ 75 ( \cot 10^0 - \cot 17^0 ) $$
