1. Suppose a Maserati is speeding down a straight highway in Nevada. For a twelve second window of time it has a constant speed of \(220\text{ ft/s}\) (\(\approx 147\text{ mph}\)). How far does the car travel in those twelve seconds?
   
   
   
   
   
   
   
2. Suppose a Maserati is speeding down a straight highway in Nevada. For a twelve second window of time it has an constant average speed of \(220\text{ ft/s}\) (\(\approx 147\text{ mph}\)). How far does the car travel in those twelve seconds?
   
   
   
   
   
   
   
3. Suppose I’m standing on the side of a Nevada freeway with a radar gun in hand. A Maserati is cruising towards me. I point my radar gun towards the car to measure its speed and, noticing this, the driver of the Maserati floors it. The car speeds past me and disappears into the distance, the radar gun losing track of the car exactly twelve seconds after I first pointed it towards them. When I started recording the car’s speed the radar gun read \(120\text{ ft/s}\) (\(\approx 80\text{ mph}\)), and when when car sped out of range twelve seconds later the radar gun read \(264\text{ ft/s}\) (\(\approx 176\text{ mph}\)). Approximately how far did the car travel in those twelve seconds?
   
   
   
   
   
   
   
4. Suppose I looked at the radar gun exactly \(6\) seconds after pointing it towards the Maserati, and saw that it reported a speed of \(174\text{ ft/s}.\) Can we use this information to get a better approximation for how far the car travelled in those twelve seconds?
   
   
   
   
   
   
   
5. Furthermore, what if we know the radar gun reported it was going \(135.75\text{ ft/s}\) after three seconds and \(221.25\text{ ft/s}\) after nine seconds? Can we use this information to get an even better approximation for how far the car travelled in those twelve seconds?
   
   
   
   
   
   
   
6. What if we could know the speed of the car at any moment during those twelve seconds? Suppose my radar gun, being quite technologically advanced, used the data it collected on the speed of the car and worked out a model \(f(t)\) for the car’s speed \(v\) at any time \(t\) during those twelves seconds: \[ v \;=\; f(t) = 120 + 2t^2 - \frac{1}{12}t^3\,. \] How far did the car travel in those twelve seconds?
   
   
   
   
   
   
   

The Full Prompt

Suppose I’m standing on the side of a Nevada freeway with a radar gun in hand. A Maserati is cruising towards me. I point my radar gun towards the car to measure its speed and, noticing this, the driver of the Maserati floors it. The car speeds past me and disappears into the distance, the radar gun losing track of the car exactly twelve seconds after I first pointed it towards them. My radar gun, being quite technologically advanced, used the data it collected on the speed of the car and worked out a model \(f(t)\) for the car’s speed \(v\) at any time \(t\) during those twelves seconds: \[ v \;=\; f(t) = 120 + 2t^2 - \frac{1}{12}t^3\,. \] How far did the car travel in those twelve seconds?