+5 A bee is on a hill looking at a building. The building is 400 feet tall. The angle of elevation from the bee to the top of the building is \(\color[rgb]{0.35,0.35,0.35}4^\circ\) and the angle of depression from the bee to the bottom of the building is \(\color[rgb]{0.35,0.35,0.35}2^\circ.\) What is the shortest distance the bee will have to fly to reach the building?
+876 Let $X$ be the position of the be.
Let $A$ be the highest point of the building.
Let $B$ be the lowest point of the building.
Let $C$ be the point that has the same altitude to $X$ (aka the shortest distance, which is what we are trying to find.)
We are given $AB = 400.$
Since we are given two angles and $AB$, we will try to relate $AB$ to $XC.$
Note $AB = AC + BC.$
We have that $AC = XC \tan 4^{\circ} \cong 0.070 XC.$
We also have that $BC = XC \tan 2^{\circ} \cong 0.035 XC.$
Thus we have $AB \cong 0.105 XC.$
And since we also have $AB = 400,$ we have that $0.105 XC \cong 400.$
$XC \cong \frac{200}{21} \cdot 400 \cong \boxed{3800 \text{ feet.}}$
