Exam-Style Question on Differentiation Optimisation

A child's play tent is made in the shape of half a cylinder. It is constructed from a fibreglass frame with material pulled tightly around it. The fibreglass frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the following diagram.

The semicircular ends each have radius \(r\) and the support rods each have length \(d\).

Let F be the total length of fibreglass used in the frame of the play tent.

(a) Write down an expression for \(F\) in terms of \(r\), \(d\) and \(\pi\).

The volume of the play tent is 0.95 m³.

(b) Write down an equation for the volume of the play tent in terms of \(r\), \(d\) and \(\pi\).

(c) Show that \(F = 2\pi r + 4r + \frac{7.6}{\pi r^2}\)

(d) Find \(\frac{dF}{dr}\)

The play tent is designed so that the length of fibreglass used in its frame is a minimum.

(e) Find the value of \(r\) for which \(F\) is a minimum.

(f) Calculate the value of \(d\) for which \(F\) is a minimum.

(g) Calculate the minimum value of \(F\).