MHB Scott's question at Yahoo Answers regarding optimization with constraint

AI Thread Summary
The discussion focuses on optimizing the area of a rectangular athletic field capped by semicircles, constrained by a 400-meter running track. The objective function for the area is defined as f(r,x)=2rx, with the perimeter constraint given by g(r,x)=2x+2πr-P=0. Through various methods, including substitution and differentiation, it is determined that the optimal dimensions occur when x equals 100 meters and r equals 100/π meters. Both the quadratic function analysis and Lagrange multipliers confirm that these values yield the maximum area. The findings indicate that the straight portions of the track are equal in length to the curved portions.
MarkFL
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Here is the question:

Scott said:
An athletic field is to be built in the shape of a rectangle x units long capped by semicircular regions of radius r at the two ends?

The field is to be bounded by a 400 meter running track. What values of x and r will give the rectangle the largest possible area?

I have posted a link there to this thread so the OP can view my work.
 
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Hello Scott:

The objective function (that which we wish to optimize) is the area of the rectangle, and is as follows:

$$f(r,x)=2rx$$

And if we let $P$ be the perimeter of the field, we may state the constraint:

$$g(r,x)=2x+2\pi r-P=0$$

Now, we have two ways to proceed. The first is to solve the constraint for one of the two variables $r$ or $x$, and then substitute into the objective function so that we have a function in one variable and then optimize by differentiation. Sol if we solve the constraint for $r$, we obtain:

$$r=\frac{P-2x}{2\pi}$$

And the substituting into the objective function, we obtain:

$$f(x)=2\left(\frac{P-2x}{2\pi}\right)x=\frac{x(P-2x)}{\pi}$$

Now, at this point we see we have a quadratic function opening downwards and we could use a precalculus technique to find the maximum value. We see the roots are:

$$x=0,\,\frac{P}{2}$$

We know then that the axis of symmetry lies midway between these roots, and so the axis of symmetry is at :

$$x=\frac{P}{4}$$

And so we have:

$$r=\frac{P-2\left(\frac{P}{4}\right)}{2\pi}=\frac{P}{4\pi}$$

Now, let's try differentiation. Recall:

$$f(x)=\frac{x(P-2x)}{\pi}=\frac{1}{\pi}\left(Px-2x^2\right)$$

Hence, differentiating with respect to $x$, and equating the result to zero to obtain the critical value(s), we obtain:

$$f'(x)=\frac{1}{\pi}\left(P-4x\right)=0\implies x=\frac{P}{4}$$

And like before, we find:

$$r=\frac{P}{4\pi}$$

We see also that:

$$f''(x)=-\frac{4}{\pi}<0$$

Since the objective function is concave down everywhere, we know the critical value is at the global maximum.

Now, let's examine a multivariable method: Lagrange multipliers. Recall, we have:

The objective function:

$$f(r,x)=2rx$$

Subject to the constraint:

$$g(r,x)=2x+2\pi r-P=0$$

Hence, we obtain the system:

$$2x=\lambda(2\pi)$$

$$2r=\lambda(2)$$

Hence, this implies:

$$\lambda=\frac{x}{\pi}=r$$

Substituting for $r$ into the constraint, we obtain:

$$2x+2\pi\left(\frac{x}{\pi}\right)-P=0$$

$$4x=P$$

$$x=\frac{P}{4}\implies r=\frac{P}{4\pi}$$

So, given that the objective function is zero for $x=0$, we may conclude that this critical point is at a maximum.

Thus, we have shown in various ways that:

$$f_{\max}=f\left(\frac{P}{4},\frac{P}{4\pi}\right)$$

Using the given value $P=400\text{ m}$, we then may conclude that the rectangular portion of the field is maximized for:

$$x=\frac{400\text{ m}}{4}=100\text{ m}$$

$$r=\frac{400\text{ m}}{4\pi}=\frac{100}{\pi}\text{ m}$$

This means that the straight portions of the track are equal in length to the curved portions.
 
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