Question

ex 4: from a given point on the ground, the angle of elevation to the top of a tree is 36.7°. from a second point, 50 feet back, the angle of elevation to the top of the tree is 22.2°. find the height of the tree to the nearest foot.

Explanation:

Step1: Set up tangent - equations

In right - triangle $ABC$, $\tan(36.7^{\circ})=\frac{h}{x}$, so $h = x\tan(36.7^{\circ})$. In right - triangle $BDC$, $\tan(22.2^{\circ})=\frac{h}{x + 50}$, so $h=(x + 50)\tan(22.2^{\circ})$.

Step2: Equate the two expressions for $h$

Since $x\tan(36.7^{\circ})=(x + 50)\tan(22.2^{\circ})$, expand the right - hand side: $x\tan(36.7^{\circ})=x\tan(22.2^{\circ})+50\tan(22.2^{\circ})$.

Step3: Solve for $x$

Rearrange the equation: $x\tan(36.7^{\circ})-x\tan(22.2^{\circ})=50\tan(22.2^{\circ})$. Factor out $x$: $x(\tan(36.7^{\circ})-\tan(22.2^{\circ}))=50\tan(22.2^{\circ})$. Then $x=\frac{50\tan(22.2^{\circ})}{\tan(36.7^{\circ})-\tan(22.2^{\circ})}$.
We know that $\tan(36.7^{\circ})\approx0.747$ and $\tan(22.2^{\circ})\approx0.408$. So $x=\frac{50\times0.408}{0.747 - 0.408}=\frac{20.4}{0.339}\approx60.18$.

Step4: Find the height $h$

Substitute $x$ into $h = x\tan(36.7^{\circ})$. $h\approx60.18\times0.747\approx45$.

Answer:

45 feet