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Donniee · Feb 14, 2012, 09:55 PM

To measure the height of the Eiffel Tower in Paris, a person stands away from the base and measures the angle of elevation to the top of the tower to be 60 degrees. Moving 210 feet closer, the angle of elevation to the top of the tower is 70 degrees. How tall is the Eiffel Tower?

corrigan · Feb 15, 2012, 12:06 AM

The tangent of an angle is defined as the proportion of the opposite side to the adjacent side of a right triangle containing that angle, also called opposite over adjacent. So we can find the height of the eiffel tower by taking the tangent of the angle of elevation and multiplying it by the distance from the eiffel tower. The problem here is that we don't know the distance from the eiffel tower, but since we have two different angles of elevation from two different distances, we can figure the rest out.

When the angle of elevation is 70 degrees we are $x$ feet away from the eiffel tower, so the height of the tower is $x \cdot \tan 70^{\circ}$ feet. Similarly, when the angle of elevation is 60 degrees we are $x + 210$ feet away from the eiffel tower, so the height of the tower is $(x +210) \cdot \tan 60^{\circ}$ feet. Since the height of the tower doesn't change, we can set these equal to each other and solve for $x$ . Then we can just multiply $x$ by $\tan 70^{\circ}$ and get the height of the tower. To solve for $x$ we have:

$x \cdot \tan 70^{\circ} = (x +210) \cdot \tan 60^{\circ}$

$\Rightarrow \frac{x+210}{x} = \frac{\tan 70^{\circ}}{\tan 60^{\circ}}$

$\Rightarrow 1+ \frac{210}{x} = \frac{\tan 70^{\circ}}{\tan 60^{\circ}}$

$\Rightarrow \frac{210}{x} = \frac{\tan 70^{\circ}}{\tan 60^{\circ}} -1$

$\Rightarrow \frac{210}{x} = \frac{\tan 70^{\circ} - \tan 60^{\circ}}{\tan 60^{\circ}}$

$\Rightarrow \frac{x}{210} = \frac{\tan 60^{\circ}}{\tan 70^{\circ} - \tan 60^{\circ}}$

$\Rightarrow x = 210 \cdot(\frac{\tan 60^{\circ}}{\tan 70^{\circ} - \tan 60^{\circ}})$

Now that we have the distance when the elevation is 70 degrees, we just multiply by the tangent of 70 degrees, and the height of the tower is:

$210 \cdot(\frac{\tan 60^{\circ}}{\tan 70^{\circ} - \tan 60^{\circ}}) \cdot \tan 70^{\circ}$

I hope this helps :)