7 The curve \(y = 15 - x ^ { 2 }\) and the isosceles triangle \(O P Q\) are shown on the diagram
The curve \(y = 15 - x ^ { 2 }\) and the isosceles triangle \(O P Q\) are shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-10_759_810_388_614}
Vertices \(P\) and \(Q\) lie on the curve such that \(Q\) lies vertically above some point ( \(q , 0\) ) The line \(P Q\) is parallel to the \(x\)-axis.
7
- Show that the area, \(A\), of the triangle \(O P Q\) is given by
$$A = 15 q - q ^ { 3 } \quad \text { for } 0 < q < c$$
where \(c\) is a constant to be found.
7 - Find the exact maximum area of triangle \(O P Q\).
Fully justify your answer.