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Calculus/Calculus rate problem

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Expert: - 3/6/2010


Question
A 50 foot tree falls such that the angle with the ground is decreasing at 5 degrees per second.(a) At what rate is the vertical distance from the tree to the ground changing when the angle is 60 degrees? (b) At what rate is the area of the right triangle changing (with the tree as the hypotenuse) at the same instant.

Answer
Convert the problem to polar coordinates.
That means y/x = tanΘ and r² = x² + y².

Take the derivative of both of these equations and put in the known variables.

It is known that the angle Θ = 60°, r = tree length = 50, x = r/2, and y = r√3.
These two equations will have two variables, dr and dΘ.

This is just a matrix, two unknowns, two equations.

Note that in the 1st equation, it is really y(t)/x(t) = tanΘ(t) and
in the 2nd equation, it is r² = x²(t) + y²(t).  The value of r is known to  be always 50'.

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