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An observer standing on the southern bank of a river sees a tree that is much more than 100 feet downstream on the northern bank of the river. The angle formed by the southern bank and the line from the observer to the tree is 15 ∘. After walking 100 feet downstream, the observer measures the angle formed by the southern bank and the line to the same tree to be 25 ∘. Find the width of the river. (can you show the work please)(also the drawing please)

Respuesta :

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Answer:

[tex]63\ \text{ft}[/tex]

Step-by-step explanation:

The distances are shown in the figure

[tex]\tan25^{\circ}=\dfrac{p}{b}\\\Rightarrow p=b\tan25^{\circ}[/tex]

[tex]\tan15^{\circ}=\dfrac{p}{b+100}\\\Rightarrow p=b\tan15^{\circ}+100\tan15^{\circ}\\\Rightarrow b\tan25^{\circ}=b\tan15^{\circ}+100\tan15^{\circ}\\\Rightarrow b=\dfrac{100\tan15^{\circ}}{\tan25^{\circ}-\tan15^{\circ}}\\\Rightarrow b=135.08\ \text{ft}[/tex]

[tex]p=b\tan25^{\circ}=135.08\times\tan25^{\circ}\\\Rightarrow p=62.99\approx 63\ \text{ft}[/tex]

The width of the river is [tex]63\ \text{ft}[/tex].

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