# Maximization problem

I've been trying to solve the following problem from Stewart's Calculus Textbook for a while without any success. My answer makes sense, but I'm looking for a way to solve it analytically. The problem concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d > r), a rope of length l is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. This happens when the distance |ED| is maximized. Show that when the system reaches equilibrium, the value of x is:

$$\frac{r}{4d}(r+\sqrt{r^2+8d^2})$$

Here is what I've done. First, I expressed |DE| as a function of x

$$|DE|(x)={a}_{2}+{a}_{3}=l-{a}_{1}+\sqrt{{r}^{2}-{x}^{2}}=l-\sqrt{{a}_{3}^2+y^2}+\sqrt{r^2-x^2}$$

from what follows that $$|DE|(x)=l-\sqrt{r^2+d^2-2xd}+\sqrt{r^2-x^2}$$ defined for $$0\leq x \leq r$$

...and it works since $$|DE|(0)=l+r-\sqrt{r^2+d^2}$$ and $$|DE|(r)=l-|r-d|$$

To find the maximum of this function, I calculated |DE|'(x)

$$|DE|'(x)=\frac{d}{\sqrt{r^2+d^2-2xd}}-\frac{x}{\sqrt{r^2-x^2}}$$

I proved the two radicals at the denominator are defined for $$0\leq x < r$$

...so basically I'm interested in finding when |DE|'(x) equals zero, more specifically the roots of

$$d\sqrt{r^2-x^2}-x\sqrt{r^2+d^2-2xd}=0$$ that becomes

$$2dx^3-(r^2+2d^2)x^2+d^2r^2=0$$

I graphed |DE|(x) and |DE|'(x) (using l = 15, r = 3, and d = 4), they are consistent with the problem. |DE|'(x) has only one root at about 2.76 and at the same point |DE|(x) has its maximum. Moreover, if you substitute my test numbers for l, r, and d in the given formula for |DE|(x) maximum you get the same numerical result.

So, how was the author of the problem able to find an analytical solution to the problem?

Thanks!

• I appreciate the diagrams, $TeX$, and work. This is a great example of how to format a question so that it will be answered. – davidlowryduda Jan 25 '12 at 1:54

## 1 Answer

Note that $x=d$ is a solution of your cubic equation (not the one you want, of course).

More generally, cubic equations that calculus students are asked to solve will almost always have at least one easy-to-guess solution (perhaps a small integer, or one of the parameters). Of course, that is not true of equations that might occur in the real world.