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A cathedral spire $200$ ft high is $250$ ft away from a church steple which is $150$ ft high. At the same instant, two crows, one from each, fly of at the same speed heading toward some grain that is on a level, straight road connecting two towers. The crows reach the grain at the same instant. How far along that road is the grain from the cathedral?

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Is this homework? If it is, we're still happy to help, just use the homework tag. Also, can you describe your thoughts on the problem so far? That will help us figure out how best to explain the problem in a way that's useful for you. – Brett Frankel Apr 30 '12 at 23:19

2 Answers 2

up vote 3 down vote accepted

Hint: To each triangle you can apply Pythagoras, and then you can go ahead and solve for $x$ and $y$.

diagram of the problem

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I like the picture! – Nick Strehlke May 2 '12 at 17:58

Draw a picture. Let $p$ be the distance of the grain from the cathedral, which is a calculus cathedral, namely a line going straight up. Let $q$ be the distance of the grain from the church. Then $q+p=200$.

By the Pythagorean Theorem, $250^2+p^2=150^2+q^2$. So we have the equations $$q+p=200,\qquad\text{and}\qquad q^2-p^2=250^2-150^2=(100)(400).$$ This can be solved pretty quickly. Hint: It is easy to find $q-p$.

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