Answer:
The height of mountain is 8491 feet.
Step-by-step explanation:
Let h be the height of the mountain ( in feet ) and x be the distance ( in feet ) of a point ( from the base of mountain ) from which the angle of elevation to the top of the mountain is 32°,
By the trigonometric ratio,
[tex]tan 32^{\circ}=\frac{h}{x}[/tex]
[tex]\implies h = x tan 32^{\circ}[/tex]
According to the question,
If the point is 1000 feet closer to the mountain then the angle of elevation is 34°,
[tex]\implies tan 34^{\circ}=\frac{h}{x-1000}[/tex]
[tex]\implies h=(x-1000) tan 34^{\circ}[/tex]
Thus, we can write,
[tex]x tan 32^{\circ}=(x-1000) tan 34^{\circ}[/tex]
[tex]xtan32^{\circ}=xtan34^{\circ}-1000tan 34^{\circ}[/tex]
[tex]x(tan 32^{\circ}-tan34^{\circ})=-1000tan 34^{\circ}[/tex]
[tex]\implies x=-\frac{1000 tan 34^{\circ}}{tan 32^{\circ}-tan34^{\circ}}[/tex]
Hence, the height of the mountain would be,
[tex]\implies h =-\frac{1000 tan 34^{\circ}}{tan 32^{\circ}-tan34^{\circ}}\times tan 32^{\circ}=8490.87006892\approx 8491\text{ feet}[/tex]
