Car problem $(40$ points$)$

A car is driving along a path described by the equation

$Y(x)=300cos(\frac{x}{1000})+300$

The car follows the path at a constant speed of $16\frac{m}{s}.$

For the interval $x=[0,1000]$ :

$($a$)$ Find the equation and plot radius of curvature of the path versus $x$

$($b$)$ Plot the path components $($tangential and normal$)$ of the car's acceleration versus $x$

When $x=700m$ :

$($c$)$ Report the car's velocity and acceleration in Cartesian coordinates

$($d$)$ Report the car's velocity and acceleration in polar coordinates

$($e$)$ Discuss some advantages or disadvantages of using each of the three coordinate systems for this

scenario. For which system was the math easier to calculate? Which might make more sense to

the driver of the car, and which might make more sense to a pedestrian standing at the location

$(x,Y)=(0,0)m?$