A $2008$ Chevy Camaro SS has a mass of $1700$ kg and it can accelerate from $0$ to $30\frac{m}{s}(67mph)$ in about $5$ seconds. A model for the velocity during this period $($in $\frac{m}{s})$ as a function of time $($in s$)$ is$,$

$v[t]=-\frac{6}{5}{t}^2+12t$

a$)$ Calculate the power $($in hp $)$ required. Note: This part does not need calculus.

b$)$ The net force is the force applied from the engine minus any resistive forces. Derive an expression for the net force as a function of time.

c$)$ Derive an expression for power vs$.$ time. What is the maximum value of power and when does it occur? $($Use calculus rather than graphing techniques to find the maximum$).$

d$)$ Use calculus to find the average value of power and verify that it matches your result from part a$.$