2. A spaceship on auto-pilot is programmed to travel according to the position function given below. However, due to a fuel leak, the ship will no longer have enough fuel to reach its programmed destination. Unfortunately, the mechanism to steer the ship manually has failed, so the only option the captain has to stop the ship from being lost in space is to turn off the engines at a time of her choosing. When the captain turns off the engines, the ship will stop accelerating. So, from that time on, it will cease to follow the given position function and, instead, will continue drifting at the velocity it was traveling at the moment the engines stopped. The captain would like the ship to drift into the nearest space station, which is located at coordinates (6,4,9). (3+ (t)=(3+t, 2+ In(t), 7 - (+ t² + 1 (a) (3 points) At what time t should the captain turn the engines off? (You may use software and/or graphical methods to help solve any equations that result from this problem. Just indicate if/when you did so.) (b) (2 points) At what speed will the spaceship be traveling when it arrives at the space station? (c) (1 point) How long after the engines are turned off will the spaceship arrive at the space station? (d) (3 points) Now, suppose the spaceship has a special backup engine designed to provide a constant "reverse thrust" that can decelerate it. Determine the magnitude of the constant deceleration that should be used - beginning at the time the main engines are turned off and ending when the ship reaches the space station - so that the ship will reach a speed of zero exactly as it arrives at the space station. (e) (1 point) If the reverse thrust you found in part (d) is used, how long after the main engines are turned off will the spaceship arrive at the space station?

Fig: 1