Question

A superhero is running toward a skyscraper to stop a villain from getting away. As he is moving, he looks up at the top of the building and his line of sight makes a 33.0° angle with the ground. After having moved 350 m, he looks at the top of the building again, and notices that his line of sight makes a 52.0° angle with the horizontal. How tall is the skyscraper?

Answer 1

$\mathrm{\theta_{1}=33^{\circ}}\rightarrow \tan\theta_{1}=\tan(33^{\circ})=0.649$

$\mathrm{\theta_{2}=52^{\circ}}\rightarrow \tan\theta_{2}=\tan(52^{\circ})=1.279$

$\mathrm{X_{1}=350m+X_{2}}$

first equation

$\tan\theta_{1}=\mathrm{\frac{h}{X_{1}}=\frac{h}{350+X_{2}}}\rightarrow$     $\mathrm{h=(350+X_{2})}\tan\theta_{1}$

second equation

$\tan\theta_{2}=\mathrm{\frac{h}{X_{2}}}\rightarrow$     $\mathrm{h=X_{2}}\tan\theta_{2}$

then,

$\mathrm{(350+X_{2})}\tan\theta_{1}=\mathrm{X_{2}}\tan\theta_{2}$

$\mathrm{X_{2}=350m}\left(\frac{\tan\theta_{1}}{\tan\theta_{2}-\tan\theta_{1}}\right)=\mathrm{360.5m}$

therefore,

$\mathrm{h=X_{2}}\tan\theta_{2}=\mathrm{(360.5m)(1.279)=461.4m}$