We are designing a track for a 200 meter race. The shape of...

We are designing a track for a 200 meter race. The shape of the track will be two $x$ -meter, long straight stretches along with two semicircular caps of radius $r,$ as shown below: (FIGURE CANNOT COPY) Inside the track, we are going to have to plant grass, pull weeds, etc. So we would like to minimize the total area of the track. What is the minimum possible area? [Hint: Find an expression for the area, and get it in terms of one variable.]

We are designing a track for a 200 meter race. The shape of the track will be two $x$ -meter, long straight stretches along with two semicircular caps of radius $r,$ as shown below:
(FIGURE CANNOT COPY)
Inside the track, we are going to have to plant grass, pull weeds, etc. So we would like to minimize the total area of the track. What is the minimum possible area? [Hint:
Find an expression for the area, and get it in terms of one variable.]

Key Concepts

Optimization

Optimization involves finding the best solution—typically a minimum or maximum—for a given problem by systematically evaluating the behavior of a function. In many real-world applications, optimization is used to improve performance, reduce costs, or achieve ideal outcomes under given conditions.

Constrained Optimization

Constrained optimization is a type of optimization where the solution must satisfy certain prescribed conditions, or constraints, in addition to optimizing a particular objective function. This requires balancing the objective function and the limitations imposed by the constraints to find the best feasible solution.

Variable Elimination

Variable elimination is a technique used to simplify problems with multiple variables by using constraint equations to express the function solely in terms of one variable. This reduction often makes the analysis, especially tasks like optimization, more tractable and easier to compute.

Differentiation

Differentiation is a method from calculus used to determine the rate at which a function changes with respect to changes in its variables. In optimization, differentiating the function and setting the derivative equal to zero helps identify critical points, which can then be analyzed to determine if they correspond to minima or maxima in the context of the problem.

Transcript

00:01 For this problem, we're looking at the dimensions of an athletic field.

00:05 So let's start by drawing a picture of what this is going to look like.

00:08 We have an athletic field that's in the shape of a rectangle, and we're told that the rectangle is x units long, and it's capped by two semicircles with a radius r at either end.

00:25 So if that's r, this distance would be 2r.

00:31 Now, for this field, we're going to have it bounded by a 400 meter racetrack.

00:37 So that is, since it's bounded, that's the perimeter.

00:41 I want the perimeter of this field to be 400 meters.

00:49 Now, the first goal is let's express the area of the rectangular portion of the field as a function of either x or y.

00:58 It doesn't matter which.

01:00 So let's see what we have.

01:01 I want the area of the rectangle.

01:07 Well, that's length times width or 2r times x.

01:12 So that's the area of my rectangle.

01:14 Now, how do r and x relate to each other? well, for that, we're going to look at the perimeter.

01:19 We're told that the perimeter is 400 meters.

01:23 And the perimeter is going to be 2x, because that's the two lengths of my rectangular field, and then two semicircles.

01:31 And each semicircle is half the circumference.

01:35 So i have a whole circle there.

01:37 That is 2 pi r.

01:41 Okay, so i just have to solve this perimeter for one of these two variables.

01:46 It doesn't matter which one.

01:48 I'm going to solve it for x...

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