A golfer launches a tee shot down a horizontal fairway; it follows a path given by $\mathbf{r}(t)=\left\langle a t,(75-0.1 a) t,-5 t^{2}+80 t\right\rangle,$ where $t \geq 0$ measures time in seconds and $\mathbf{r}$ has units of feet. The $y$ -axis points straight down the fairway and the $z$ -axis points vertically upward. The parameter $a$ is the slice factor that determines how much the shot deviates from a straight path down the fairway.
a. With no slice $(a=0),$ sketch and describe the shot. How far does the ball travel horizontally (the distance between the point the ball leaves the ground and the point where it first strikes the ground)?
b. With a slice $(a=0.2),$ sketch and describe the shot. How far does the ball travel horizontally?
c. How far does the ball travel horizontally with $a=2.5 ?$