Optimise 2D composite shape

A question is this type if and only if it requires expressing the perimeter or area of a 2D composite shape (rectangle joined to semicircle, etc.) in terms of one variable using a constraint, then using calculus to find the minimum or maximum value.

2 questions · Standard +0.3

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Edexcel C12 2019 June Q15
11 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-44_537_679_258_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram not drawn to scale Figure 3 shows the plan view of a garden. The shape of this garden consists of a rectangle joined to a semicircle. The rectangle has length \(x\) metres and width \(y\) metres.
The area of the garden is \(100 \mathrm {~m} ^ { 2 }\).
  1. Show that the perimeter, \(P\) metres, of the garden is given by $$P = \frac { 1 } { 4 } x ( 4 + \pi ) + \frac { 200 } { x } \quad x > 0$$
  2. Use calculus to find the exact value of \(x\) for which the perimeter of the garden is a minimum.
  3. Justify that the value of \(x\) found in part (b) gives a minimum value for \(P\).
  4. Find the minimum perimeter of the garden, giving your answer in metres to one decimal place.
Edexcel C2 2005 January Q9
12 marks Standard +0.3
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-16_821_958_301_516}
\end{figure} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is \(2 x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2 x\) metres. The perimeter of the stage is 80 m .
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 } .$$
  2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
  3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
  4. Calculate, to the nearest \(\mathrm { m } ^ { 2 }\), the maximum area of the stage.