[问答题]
A town is planning to const
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tf 0tuv qfj97zxy) zsk558al (uct a jogging path in a grass field 170 $\mathrm{~m} $ long and 70$\mathrm{~m}$ wide. The path is to be the shape of a rectangle with two semicircles of radius x , as shown in the diagram. The sides of the rectangle connecting the circles are to be 100$ \mathrm{~m}$ long.
1. Write down a function, P , (in metres) for the perimeter of the jogging path, in terms of the radius, x .
2. Determine the domain and range of P , taking into consideration the dimensions of the grass field.
3. Find an equation for the inverse function $P^{-1}(x)$ . Express your answer in the form $ P^{-1}(x)=m x+c$ .
The designers of the path are deciding whether the total length of the path should be 300 $\mathrm{~m}$, 400 $\mathrm{~m}$ , or 500 $\mathrm{~m}$ . The designers want to maximise the perimeter of the path, but fit the path in the grass field.
4. Determine which length is most suitable, given the dimensions of the grass field.
