Center or Mass with Optimization

Question

I was given the following problem by my Multivariable Calculus Professor. However, it seems that I have hit a wall with being able to solve it.

1. Consider the “ice-cream-cone”-shaped solid that is described by the following inequalities in spherical coordinates: 0 ≤ ρ ≤ a, 0 ≤ φ ≤ π/3, 0 ≤ θ ≤ 2π. Find the center of mass of this solid, assuming that the density at each point is proportional to that point’s distance to the origin.

For this, I obtained $$(0,0,\frac{4}{5})$$

2. (Extra Credit) Suppose we change the bounds on φ to be 0 ≤ φ ≤ α for some angle α. Now what is the center of mass? What is the largest possible value of the z-coordinate of the center of mass?

However, for this particular part, I ended up obtaining

$$(0,0, \frac{4 \left(a \sin ^2(\alpha \right)}{5 \sin \left(\frac{\alpha }{2}\right)}$$

I know that in order to find the max value of the function, I'd have to take the derivative, set the derivative to 0 and then solve for alpha.

However, after taking the first derivative, I am able to simplify at most as $$\frac{\sin (\alpha ) (a-\sin (\alpha ))}{10 \sin ^3\left(\frac{\alpha }{2}\right)}$$

How can I better approach this problem?

Answer 1

You methodology is correct. If the term $$\frac{\sin (\alpha ) (a-\sin (\alpha ))}{10 \sin ^3\left(\frac{\alpha }{2}\right)}$$ is correct (I have not done the computations myself), then it is easy to solve for $\alpha$:

• either $\sin \alpha =0$, i.e., $\alpha=n\pi, n\in \mathbb{Z}$
• either $a-\sin \alpha =0$, i.e., $\alpha=\sin^{-1}a+2n\pi, n\in \mathbb{Z}$

It is not difficult to see that the first case leeds to a minimum (the cone is "flat"), while the second is a maximum.