Optimization Approach Using Calculus

Again assuming we have an angle of elevation of Pi/4 radians or 45 degrees we determine the range (rangeHill(k)) when shooting from the hill as a function of how far back we shoot, i.e. the catapult is located at (-k, Sqrt[3] k).

The rangeHill function determines the range as a function of k (rangeHill(k)) by substituting the time the projectile strikes the ground as a function of k into the x[t,k] function.

We plot this rangeHill function and estimate its maximum range to be at around 3,000 feet when the catapult is moved back about 900 ft.

But what is the exact value of k which will permit the projectile to hit the target at (3000, 0)?

Thus if we move to location (-459.358, Sqrt[3] 459.358) we can hit the target exactly. This is a distance of 918.716 ft.

This says that if we move the catapult to a point 459.358 feet back from the origin and 795.632 feet up the hill we can hit the target at (3000, 0) with our projectile.