Maggie

Maggie observes a car and a tree from a window. The angle of depression of the car is 45 degrees, and that of the tree is 30 degrees. If the distance between the vehicle and the tree is 100 m, find Maggie's distance from the tree.

Correct answer:

t =  236.6025 m

Step-by-step explanation:

$$\begin{gathered} \alpha =45{}^{\circ } \\ \beta =30{}^{\circ } \\ d=100\text{m} \\ \tan \alpha =h:x \\ \tan \beta =h:t=h:(x+d) \\ h=x\cdot \tan \alpha \\ h=(x+d)\cdot \tan \beta \\ x\cdot \tan \alpha =(x+d)\cdot \tan \beta \\ x\cdot \tan \alpha =x\cdot \tan \beta +d\cdot \tan \beta \\ x\cdot \tan \alpha -x\cdot \tan \beta =d\cdot \tan \beta \\ x=d\cdot \frac{\tan \beta }{\tan \alpha -\tan \beta }=d\cdot \frac{\tan 30°}{\tan 45°-\tan 30°}=100\cdot \frac{\tan 30°}{\tan 45°-\tan 30°}=100\cdot \frac{0.57735}{1-0.57735}=136.60254\text{m} \\ t=x+d=136.6025+100=236.6025\text{m} \end{gathered}$$

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