15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-46_396_591_251_664}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a plan for a garden.
The garden consists of two identical rectangles of width \(y \mathrm {~m}\) and length \(x \mathrm {~m}\), joined to a sector of a circle with radius \(x \mathrm {~m}\) and angle 0.8 radians, as shown in Figure 2.
The area of the garden is \(60 \mathrm {~m} ^ { 2 }\).
- Show that the perimeter, \(P \mathrm {~m}\), of the garden is given by
$$P = 2 x + \frac { 120 } { x }$$
- Use calculus to find the exact minimum value for \(P\), giving your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
- Justify that the value of \(P\) found in part (b) is the minimum.
\includegraphics[max width=\textwidth, alt={}, center]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-49_83_59_2636_1886}