Problem

The diagram shows the curve with equation $y=4x^2-x^3$ and the tangent to the curve at the point $P$. The point $P$ has $x$-coordinate $3$.

(a) Find the equation of the tangent to the curve at the point $P$. Give your answer in the form $y=mx+c$.

(b) The shaded region is bounded by the curve, the $x$-axis and the tangent to the curve at $P$.

Find the exact area of the shaded region.

The graph of $y=4x^2-x^3$ is transformed by a stretch of scale factor $\frac{1}{3}$ in the $x$-direction. The point $Q$ is the image of $P$ under this transformation. The transformed shaded region is bounded by the transformed curve, the $x$-axis and the tangent to the transformed curve at $Q$.

(c) (i) Find the equation of the transformed curve in the form $y=mx^2+nx^3$, where $m$ and $n$ are integers to be found.

(ii) State the coordinates of $Q$ and the area of the transformed shaded region.