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Calculus for Scientists and Engineers: Early Transcendental

Where do they meet? Kelly started at noon $(t=0)$ riding a bike from Niwot to Berthoud, a distance of $20 \mathrm{km},$ with velocity $v(t)=15 /(t+1)^{2}$ (decreasing because of fatigue). Sandy started at noon $(t=0)$ riding a bike in the opposite direction from Berthoud to Niwot with velocity $u(t)=20 /(t+1)^{2}$ (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.

a. Make a graph of Kelly's distance from Niwot as a function of time.

b. Make a graph of Sandy's distance from Berthoud as a function of time.

c. How far has each person traveled when they meet? When do they meet?

d. More generally, if the riders' speeds are $v(t)=A /(t+1)^{2}$ and $u(t)=B /(t+1)^{2}$ and the distance between the towns is $D,$ what conditions on $A, B,$ and $D$ must be met to ensure that the riders will pass each other?

e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).