$(15$ points$)$ A particle moves along a path defined by the line $y=x^2-45($meters$).$ The horizontal position of the particle is defined by the equation $x=5t($meters$).$

a$.$ Determine the position coordinates $(x,y)$ at a time of $3$ seconds.

b$.$ Determine the magnitude of the velocity of the particle at a time of $3$ seconds.

c$.$ Determine the magnitude of the acceleration of the particle at a time of $3$ seconds.

d$.$ Explain the difference in how the solution is presented for parts a and b $.$