## Exam-Style Question on Differentiation Optimisation## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 104. This question is similar to one that appeared on an IB Studies paper in 2014. The use of a calculator is allowed.

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 ex

The volume of the play tent is 0.95 m^{3}.

(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 \(F\).

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