4200.27 – The Top of the Turbine

From a point on the ground 252 meters from the wind turbine tower, the top of the turbine has an angle of elevation that doubles as you move 172 meters closer to the base of the tower. What is the height of the tower in meters?

Solution

Referring to the diagram below,

\begin{aligned} \tan(x) &=& \frac{h}{252}, \\ \tan (2x) &=& \frac{h}{80} \\ &=& \frac{2 \tan(x)}{1 - (\tan(x))^2} \\ &=& \frac{\frac{2h}{252}}{1 - \frac{h^2}{252^2}} \end{aligned}

More algebra gives us,

\begin{aligned} \frac{h}{80} &=& \frac{\frac{2h}{252}}{1 - \frac{h^2}{252^2}}, \\ 252^2 - h^2 &=& 160 \cdot 252, \\ h^2 &=& 252^2 - 160 \cdot 252 \\ &=& 23,814. \end{aligned}

It follows that $h \approx 152 \text{ meters}.$

That's a big one. As of November 2017 the tallest one in the world was in Germany, at 178 meters. You could check Google to see whether it's still the tallest.