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Julio Ernesto Argueta
5 months ago

COULD IT WORK WITH TANGENT

Matheno
Editor
5 months ago

$\tan \theta = \frac{y}{x}$
instead of using $\cos \theta = x/10$ as we did above?

Assuming that right: great question, and we love that you’re thinking this through!

Writing $\tan \theta = y/x$ in Step 2 would be a correct representation of the situation given the figure we drew in Step 1. So up to that point, yes.

However, you’d run into a problem in the following steps: When you take the derivative in Step 3, because both x and y are changing as the ladder slides down the wall, you would have both $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ terms. And then when you come to Step 4, we don’t know the value of that latter quantity, so you’d be stuck. (Note also that the derivative would be much harder to take in Step 3, since both x and y are variables and so you’d have to use the quotient rule. By contrast, when using $\cos \theta$ the 10 in the denominator on the right is a constant and so the derivative is quite simple to write down.)

So in short: as a practical matter, no, you can’t use $\tan \theta$ to arrive at the solution here. (At least not nearly as easily. You actually find $\dfrac{dy}{dt}$ as its own problem – see “How fast is the ladder’s top sliding?” But then you’re solving an entirely different problem as another sub-part of this problem when it’s not necessary.)

But we want to emphasize that starting your solution attempt with $\tan \theta = y/x$ is perfectly reasonable. It’s only through experience (and heading down a lot of ultimately-unproductive paths) while we were learning this stuff ourselves that we now know more quickly which the best approach to take is. So: keep practicing!!