A boundary case in a stadium model
8 written marks
Extension: boundary cases in modelling.
A stadium consists of a rectangle of length \(\ell>0\) and width \(2r\), with an external semicircle of radius \(r>0\) on each short side. Its outside perimeter is \(48\) cm. No differentiation is to be used.
Part a · 2 marks
Express \(\ell\) in terms of \(r\) and give the full permitted interval for \(r\).
Part b · 3 marks
Show that its area can be written as \(\displaystyle A=\frac{576}{\pi}-\pi(r-\frac{24}{\pi})^2\).
Part c · 3 marks
Explain whether a largest area is attained when \(\ell>0\). Then state the largest area and shape if the condition is relaxed to \(\ell\ge0\).
Complete-answer checking, worked solutions and indicative written criteria.
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