15.
\begin{figure}[h]
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\caption{Figure 3}
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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 \(100 \mathrm {~m} ^ { 2 }\).
- Show that the perimeter, \(P\) metres, of the garden is given by
$$P = \frac { 1 } { 4 } x ( 4 + \pi ) + \frac { 200 } { x } \quad x > 0$$
- Use calculus to find the exact value of \(x\) for which the perimeter of the garden is a minimum.
- Justify that the value of \(x\) found in part (b) gives a minimum value for \(P\).
- Find the minimum perimeter of the garden, giving your answer in metres to one decimal place.