maine law of sines An observer who is on the ground upon sighting
 the peak of a hill notes that it is 30 ° the angle that the straight line
 linking your foot to the peak of the hill makes the so-called flat ground.
 He walks 300 m towards the hill and makes a new measurement,
 finding now 45 °.  Based on this situation, judge the
 following items.
If R is the distance, in meters, from the observer's foot to the peak of the hill in the first measurement and if r is the same distance in the second measurement, then R 2r.
Note:the question is if R = 2r.

I want to know if my calculation is correct
 A: Good,I believe it's twice as much.
A: Let $h$ be the height of the hill. Then,
$$R = h\csc30^\circ,\>\>\>\>\> r = h\csc45^\circ$$
whose ratio leads to,
$$\frac R{r} = \frac{\csc30^\circ}{\csc45^\circ} = \sqrt2$$
Thus, $R\ne2r$. (Note the distance 300m is irrelevant.)
A: I don't think that's correct. Using the law of sines, as you appear to have done, you get
$$\frac{R}{\sin 90^{\circ}} = \frac{x+300}{\sin 60^{\circ}},$$
not (as you wrote) $\frac{300}{\sin 60^{\circ}}$.
If you want to use the law of sines, concentrate on the triangle whose sides are $r$, $R$, and $300$, so that
$$\frac{r}{\sin 30^{\circ}} = \frac{R}{\sin 135^{\circ}}\Rightarrow \frac{r}{1/2} = \frac{R}{\sqrt{2}/2}\Rightarrow R = \sqrt{2}\,r.$$
Alternatively, you could note that $x$ is also the height of the hill, so that
$$\frac{x}{r} = \cos 45^{\circ},\quad \frac{x}{R} = \cos 60^{\circ}$$
and proceed from there.
A: Find their ratio for comparison:
$$\dfrac {R}{r}= \dfrac {h/ \sin 30^{\circ}}{h/ \sin 45^{\circ}}$$
$$ = \sqrt{2}, \ne 2. $$
For this ratio ( of linear dimensioned) height & horizontal distance magnitudes play no role.
