A person starts out 17 miles from the base of a tall mountain, and looks up at a 4 degree angle of elevation to the top of the mountain. When they move 12 miles closer to the base of the mountain, what will be their angle of elevation when they look to the top?

We have been given that a person starts out 17 miles from the base of a tall mountain and looks up at a 4 degree angle of elevation to the top of the mountain. We are asked to find the measure of angle of elevation when they move 12 miles closer to the base of the mountain.

We can see that height (h) of the mountain is will be equal to both angles.

Let us find expression height of mountain.

Since tangent relates opposite side of right triangle to adjacent, so we can set an equation as:

[tex]\text{tan}(4)=\frac{h}{17}[/tex]

[tex]17\text{tan}(4)=h[/tex]

Similarly, we can set another equation for h.

[tex]\text{tan}(x)=\frac{h}{(17-12)}[/tex]

[tex]\text{tan}(x)=\frac{h}{5}[/tex]

[tex]5\text{tan}(x)=h[/tex]

so we can equate both expressions as:

[tex]17\text{tan}(4)=5\text{tan}(x)[/tex]

[tex]17(0.069926811944)=5\text{tan}(x)[/tex]

[tex]1.188755803048=5\text{tan}(x)[/tex]

[tex]\text{tan}(x)=\frac{1.188755803048}{5}[/tex]

[tex]\text{tan}(x)=0.2377511606096[/tex]

[tex]x=\text{tan}^{-1}(0.2377511606096)[/tex]

[tex]x=13.373839748235^{\circ}\approx 13.4^{\circ}[/tex]

Therefore, the angle of elevation would be 13.4 degrees, when they move 12 miles closer to the base of the mountain.