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The diagram shows a metal plate consisting of a rectangle with sides \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and a quarter-circle of radius \(x \mathrm {~cm}\). The perimeter of the plate is 60 cm .
- Express \(y\) in terms of \(x\).
- Show that the area of the plate, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 30 x - x ^ { 2 }\).
Given that \(x\) can vary,
- find the value of \(x\) at which \(A\) is stationary,
- find this stationary value of \(A\), and determine whether it is a maximum or a minimum value.