4200.21 – A New Volcano

Suppose a new volcano suddenly appeared where your school used to be. You’d want to know how tall it was, wouldn’t you. And so you might set out from your house towards the volcano and take sightings along the way. Suppose, then, that you stop a safe distance (point A in the figure) and measure an angle of elevation to be 21 degrees. After marching resolutely another 500 meters to B you find the elevation has increased to 35 degrees. Further you dare not go as there is lava still flowing. Time to retreat and calculate! What do you tell the world is the height of the new volcano?

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Solution

In the figure, we first fill in all the angles. Then we can get VB using the Law of Sines, and then get hh from \triangle VBC. First,

500sin⁡(14)=VBsin⁡(21)VB=500sin⁡(21)sin⁡(14)≈740meters \begin{aligned} \frac{500}{\sin(14)} &=& \frac{VB}{\sin(21)} \\ VB &=& \frac{500 \sin(21)}{\sin(14)} \\ &\approx& 740 \text{meters} \end{aligned}

Next we find hh:

h=sin⁡(35)⋅740≈424metersh = \sin(35) \cdot 740 \approx 424 \text{meters}

That’s a mighty tall volcano to make a sudden appearance! (or is it?)

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