Answer:
0.68 miles
Step-by-step explanation:
Let's say that the height of the mountain is h
The distance between the mountain and the car, after moving 23 miles closer to the mountain, will be x (let's pose it as x for now)
tan(1.5°) = opposite/adjacent = h/23 + a
=> h = (23 + a)tan(1.5°) ---- (1)
tan(13°) = h/a
=> h = a(tan(13°)) ---- (2)
Now since h is common among the two equations, we can equate them;
[tex]\left(23+a\right)\tan \left(1.5^{\circ \:}\right)=a\left(\tan \left(13^{\circ \:}\right)\right),\\23\tan \left(1.5^{\circ \:}\right)+\tan \left(1.5^{\circ \:}\right)a=\tan \left(13^{\circ \:}\right)a,\\\\\tan \left(1.5^{\circ \:}\right)a-\tan \left(13^{\circ \:}\right)a=-23\tan \left(1.5^{\circ \:}\right),\\\left(\tan \left(1.5^{\circ \:}\right)-\tan \left(13^{\circ \:}\right)\right)a=-23\tan \left(1.5^{\circ \:}\right),\\\\[/tex]
[tex]a=\frac{23\tan \left(1.5^{\circ \:}\right)}{\tan \left(13^{\circ \:}\right)-\tan \left(1.5^{\circ \:}\right)} = 2.94249...[/tex]
[tex]h = 2.94249tan\left(13^{\circ }\right) = 0.67932\dots[/tex]
The height of the mountain ≈ 0.68 miles
