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\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817}
The diagram shows a glass window consisting of a rectangle of height \(h \mathrm {~m}\) and width \(2 r \mathrm {~m}\) and a semicircle of radius \(r \mathrm {~m}\). The perimeter of the window is 8 m .
- Express \(h\) in terms of \(r\).
- Show that the area of the window, \(A \mathrm {~m} ^ { 2 }\), is given by
$$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$
Given that \(r\) can vary,
- find the value of \(r\) for which \(A\) has a stationary value,
- determine whether this stationary value is a maximum or a minimum.