Differentiation

An industrial laser cutter follows a trajectory $C\,$ defined by the parametric equations

2p^2, \quad y = \frac{2}{3}p^3 + 4p^2 - 14p + k

where $k\,$ is a constant and $\neq 0$ .

Find $dydx\displaystyle \frac{dy}{dx}$ in terms of $p$ .

[2]

The line $l\,$ is the normal to the curve $C\,$ at the point $A\,$ where $p = 1$ .

Given that the tangent to $C\,$ at the point $B\,$ is parallel to $l$ ,

show that the parameter $p\,$ at point $B\,$ is a solution of the equation

p^2 + 2p - 7 = 0

[4]

Hence find the value of $p\,$ at $B$ , justifying your choice given that the $x$ -coordinate of $B\,$ is greater than 10.

[3]

Given that the $y$ -intercept of $l\,$ is $23\displaystyle \frac{2}{3}$ ,

determine the value of $k$ .

[3]