How Can I Optimize a Racecourse Using Similar Triangles and Cone Volume?

[TheNormalForc]

I know how to optimize, but I'm having trouble starting both. I think the trick to the first one is something to do with similar triangles, but I can't quite articulate an equation to do the problem with. I know the second one requires the volume, and I think the area of a circle or the surface area of a cone might come in handy. I'm fully capable of deriving, and determining the max/minimums. But the second part of one has me stumped throughly.

I can't demand an answer, and I don't really want to be told one. But I implore you very generous people to expediate your help, as I've only been allowed a few hours to do these.

 

[lippyka]

To start the first problem, you can begin by setting up a system of equations. You are given the sides of two triangles, and can use that to calculate the angles using the Law of Cosines. From there, you can use similar triangles to determine the length of the other side of the second triangle. Once you have this information, you can use the Pythagorean theorem to calculate the length of the third side of the second triangle. From here, you can calculate the area of the two triangles and find the difference between them. To start the second problem, first use the volume of a cone formula to calculate the volume of the cone. From there, you can set up an equation in terms of x, representing the radius of the cone. This equation will give you the maximum and minimum values of x, which you can then use to calculate the maximum and minimum volumes of the cone.