A 400m race track is to be built with 2 straightaways and 2 semicircle's at the end. The straightaways cannot be shorter than 100m. What radius of the semicircle can produce the maximum total area of the track?
It's a question from beginners calculus. Teacher usually uses method of finding perimeter equation, finding surface area equation, substitution, first derivative, etc. Usually wouldn't have that much trouble with optimization problems, but since the "no less than 100m" was thrown in I have no idea where to start. Any help would be greatly appreciated!
First, we have to interpret the problem as intended. This requires some inferences about the intent in this case since the author did not describe the track in the way that a track is described in real life.
In real life a $400$ meter running track would be constructed between two ovals, an inner oval of perimeter $400$ meters and a larger outer oval. I would consider the area of the track to be the area between the ovals.
But that real-life interpretation would make the problem unanswerable, so we must assume the author does not understand how to describe a race track to a runner, but has some simplified problem in mind instead. So we assume that there is only one oval to consider, that this oval has perimeter $400$ meters, and that the "area of the track" is the total area enclosed by this oval.
You can express the area of the track either as a function of the radius $r$ of the straight length $L$. Then you can minimize the value of that function within the interval in which the chosen variable "lives". For example, if you choose to express the area of the track as a function of $L$, you know you are looking within the interval $[100,\infty)$ because you were told that the length of the straight segment has to be at least $100.$ You can also figure out a maximum value of $L$ so that you are actually looking only in a finite interval.
If you choose to write the area as a function of $r$ then you must figure out the minimum and maximum values of $r.$ The minimum value of $L$ is a clue.
Once you have your minimum and maximum values of your variable, you find the maximum of the function on the interval between those two values.
How you approach this would depend on what you have been taught about maximization of a function within an interval. Since this problem makes no sense in real life, it can only be from a textbook or other educational setting, and it seems very unlikely that your textbook or teacher would expect you to be able to solve it with no instruction at all on what to do about the boundedness of the interval. Look over the book and/or course notes to see what you are expected to know. If you don't understand the instructions, edit the question to tell us what they are and someone may be able to explain it better.