Barbie has entered a bicycle race. Moving at a constant velocity meters per second, she passes a water station. At that instant ( = 0 seconds), her support car starts from the water station to accelerate after her, beginning from a dead stop. Suppose the distance travelled by Barbie in seconds is given by the function () = and

distance travelled by the support car is given by the function () = 1 (102 3), where distance is measured 3

in meters. This latter function is carefully calculated by her crew so that at the instant the car catches up to the racer, their velocities will match. At that moment, Ken will hand Barbie a cold water bottle and the car will immediately fall behind.

(a) HowfastisBarbietravelling?

(b) How long does it take for the support car to catch up to Barbie?

(c) At what distance from the refreshment station does the support car meet up with Barbie?

(d) Now suppose that Barbie is riding at a constant velocity , which may be different than the value found in part (a). Find an expression for the times when the car and the bike meet which gives these times in terms of her velocity . (You will need to use the Quadratic Formula.) How many times (after = 0) would the car and the bike meet if Barbie were going slower than the velocity found in part (a)? Faster?

(e) Consider a pair of axes with time measured horizontally and distance vertically. Use an online graphing tool (I recommend https://www.desmos.com/calculator) to create graphs that depict the distance

travelled by Barbie () = and by the car () = 1 (102 3) plotted on the same axes for the 3

three different possibilities for the velocity discussed in this project. You will have three graphs: one for the bike's velocity found in part (a), one for a slower bike, and one for a faster bike. (Each of your three graphs will have the same () graphed with a different (), depending on the value you use for .) If Barbie had been going any slower or faster than the velocity you found in part (a), passing the drink would not have been so easy. Why? Justify your answer.