Maths involving differentiation, perimeter and area. Struggling

Hello all, I have been attempting to work through this question set and I am stuck at the moment.
My answer for 1a is Perimeter = 2x +2pir. I’m not to sure what is meant by the relationship between x and r part though
Anyone here able to guide me through the correct steps?
Thanks, really appreciate any help.
Some planners are designing a sports field. Their idea is to have an 800m running track, with the space in the middle to be used for other sports events. The design has the running track composed of two parallel straight portions and two semi-circular ends. The perimeter of the inside lane of the track must be 800 metres.
1a) If the length of one of the straight portions of the track is x and the radius of each of the semi-circular ends is r , find an expression for the perimeter of the (inside lane of the) track, and hence find a relationship between x and r.

1b) The rectangular area enclosed by the straight portions of the track and the diameters of the end semicircles is to be used for other sports. Thus, the designers want to make this rectangle as large as possible whilst keeping the perimeter of the track fixed at 800m. Find an expression for the area of this rectangle, firstly in terms of x and r , then in terms of x only.
1c) Use differentiation to find the values of x and r which gives the area of this rectangle a stationary value, and verify that the stationary value is a (local) maximum of the area.

Hi Liam,

You have started well, but you are not using all of the information you are given, and that;s why you are stuck.

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1(a)
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Perimeter
= 2*{rectangle side} + 2*{semicircle perimeter}.
= 2 * x + 2 * π r.
= 2 (x + πr).

But, the perimeter must be 800m. Hence:
800 = 2 (x + πr).

And so:
400 = x + πr.

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1(b)
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Rectangular area
= {rectangle length} * (2 * {radius of semicircle}).