Optimization, rectangle inscribed inside arch of the curve.

Question

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area?

Attempt: I have graph it and by looking at the graph, I can see the area of the inscribed rectangle is $A = (2x)(y)$. Where $2x$ is the total base, and y is the height.

Now I substitute $y = 4\cos(0.5x)$ on the $A = (2x)(y) = 2x[4\cos(0.5x)]$.

So $A = 8x\cos(0.5x)$. Then the derivative using the product rule is with respect to $x$ is

$$A' = 4\cos(x) - 4x\sin(0.5x)$$

However, when I tried to set the derivative equal to zero to find the critical numbers I don't know how to solve $\cos(x) = x\sin(0.5x)$. Can anyone please help me? I would really appreciate the help. Thank you.

Answer 1

Let $y = 0.5x$ Then you have $cos(2y) = 2y\ sin(y)$. Now try a double angle identity, and perhaps quadratic equation. Otherwise, you can solve it numerically, at worst case.

Answer 2

First, I think that there is a small mistake in the derivative since $$A = (2x)(y) = 2x[4\cos(0.5x)]=8 x \cos(\frac{x}{2})$$ lead to $$A'=8 \cos \left(\frac{x}{2}\right)-4 x \sin \left(\frac{x}{2}\right)$$ you want to be zero. Then, as you notice, the problem is to solve the equation $$f(x)=2 \cos \left(\frac{x}{2}\right)- x \sin \left(\frac{x}{2}\right)=0$$ which could be written in many other forms.

However, this class of equations, which involve algebraic and trigonometric functions cannot be solved analytically and numerical methods must be used.

If you plot $f(x)$ between $0$ and $\pi$, you will notice that there is a solution somewhere close to $2$. So, let us use Newton iterative method which write $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ and let us start at $x_0=2$. The successive iterates will be $1.72907$, $1.72068$, $1.72067$ which is the solution for six significant figures.