Answer:
[tex]844\:\text{ft}[/tex]
Step-by-step explanation:
I've created a diagram and added steps on GeoGebra for reference. Please refer to the diagram below:
In any right triangle, the tangent of an angle is equal to its opposite side divided by its adjacent side (o/a).
Therefore, we have the two equations:
[tex]\begin{cases}\tan 14^{\circ}=\frac{h}{819}\\\tan 7^{\circ}=\frac{h}{819+x}\end{cases}[/tex]
Because [tex]h=h[/tex] (reflexive property), we have:
[tex]819\tan 14^{\circ}=\tan 7^{\circ}(819+x)[/tex]
Isolating and solving for [tex]x[/tex]:
[tex]\frac{819+x}{819}=\frac{\tan 14^{\circ}}{\tan 7^{\circ}},\\\\1+\frac{x}{819}=\frac{\tan 14^{\circ}}{\tan 7^{\circ}},\\\\\frac{x}{819}=\frac{\tan 14^{\circ}}{\tan 7^{\circ}}-1,\\\\x=819(\tan 14^{\circ}\cot 7^{\circ}-1),\\\\x=844.07256243\approx \boxed{844\text{ ft}}[/tex]
