We have a cannon that can fire projectiles with speed var$$ at an angle $.$ What angle should we use to force our cannon to hit a target that is $R$ meters away. The trajectory of the projectile is given by the equation:

$y(x)=xtan-\frac{1}{2}g(\frac{x}{\text{v}\text{a}\text{r}cos}{)}^2$

The distance $R$ is the location of x when y is zero. So the function to determine the angle given a distance $R$ is:

$f()=$Rtan$-\frac{1}{2}g(\frac{R}{\text{v}\text{a}\text{r}cos}{)}^2$

Where var$$ is the velocity $(\frac{m}{s}),R$ is the target range $(m),$ is the angle needed to reach the target range $($degrees$),$ and $g$ is the gravity $(\frac{m}{s^2}).$

What is the angle $$ needed to reach a target $150$ meters away if the speed of your projectile is $60\frac{m}{s}?$

Task $1$: Due $11/06$ at $11$:$59$ pm

Plot the function to determine the initial interval for closed methods. Use an interval of width $4$ degrees. In addition, use the plot to determine the initial guess for open methods. Use the degree nearest the root. For secant do $1$ degree less than the initial guess for Newton $-$Raphson for $x_{-1}.$ Display interval and initial guesses with fprintf.

Using a tolerance of ${10}^{-8}.$ Plot the Absolute Percentage Relative Error vs$.$ iteration number for each iteration for all $4$ methods on the same plot. There is a plot in lecture notes you can use as a reference. Include title, axis labels and a legend.

For tolerances ${10}^{-4},{10}^{-6}$ and ${10}^{-8}$ report how many iterations of each method are required to reach the tolerances.

Discuss which methods are better. Weigh computational effort alongside coding effort.

For $2-4$ write your root finding methods as user defined functions at the bottom of your script and call the functions to perform the relevant tasks.