When a fighter jet travels faster than sound, it generates a sonic boom shock wave in the shape of a cone. If the plane is flying at a constant altitude, this cone intersects the ground in the shape of one branch of a hyperbola. Suppose a jet is flying north at level altitude and at a speed of 500 m/s (which is a supersonic speed). In this model, let the position of the jet be the origin. The vertex of the sonic boom hyperbola is on the ground 15 km behind the jet, and the hyperbola has an eccentricity of 1.5. Use a scale of 1 unit = 1 km for this model. Answer the following:
I) Find the coordinates of the focus and the equations of the asymptotes. Round all answers to the nearest hundredth.
II) Write an equation for the sonic boom hyperbola in standard form.
III) A person on the ground will hear the sonic boom when the hyperbola passes over him. Suppose the jet is located at the otigin at time t = 0. The jet and the sonic boom line are moving due north at 500 m/s, and John is standing at (7,10). Calculate the time, in seconds, until John will hear the sonic boom. Round the answer to the nearest tenth of a second.