Question: Let \(y\) be the function given by \(f(x)=9 x-x^{3}\), and let \(t\) be the line \(y=-3 x+16\), where \(t\) is tangent to the graph of \(f\) Let \(P\) be the region bounded by the graph of \(f\) and the \(x\) -axis, and let \(\mathrm{Q}\) be the region bounded by the graph of \(f\), the line \(t\), and the \(x\) -axis, as shown below.

1. Use methods of calculus to show that the point at which line \(t\) is tangent to \(y=f(x)\) has coordinates \((2,10)\).
2. Find the area of region \(\mathrm{P}\).
3. Set up but do not integrate an integral expression in terms of one variable for the area of region \(\mathrm{Q}\).
4. Set up but do not integrate an integral expression in terms of one variable for the volume of the solid generated when region \(\mathrm{Q}\) is revolved about the \(x\) -axis.

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