Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 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.

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

Accepted Answer

Define the position functions for Kelly and Sandy based on their velocities and starting points. Since Kelly starts at Niwot (position 0) and travels toward Berthoud (position 20 km), her position at time $$t is $$given by integrating her velocity $$v(t) = \frac{15}{(t+1)^2$}$$ from 0 to $$t$$:

$$ K(t) = \int_0^t \frac{15}{(s+1)^2} \, ds $$ Similarly, Sandy starts at Berthoud (position 20 km) and travels toward Niwot (position 0 km) with velocity $$u(t) = \frac{20}{(t+1)^2$}. $$Since she moves in the opposite direction, her position at time $$t is$$:

$$ S(t) = 20 - \int_0^t \frac{20}{(s+1)^2} \, ds $$ Calculate the integrals for both $$K(t)$$ and $$S(t) to $$express their positions explicitly as functions of $$t. $$Recall that

$$ \int \frac{1}{(s+1)^2} ds = -\frac{1}{s+1} + C $$ Set the positions equal to find the meeting time $$t_m$$:

$$ K(t_m) = S(t_m) $$

This equation means they are at the same point on the 20 km path at time $$t_m. $$Solve the equation from step 4 for $$t_m. $$Once $$t_m is $$found, substitute it back into $$K(t)$$ and $$S(t) to $$find how far each person has traveled when they meet.

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Key Concepts

Position as an Integral of Velocity

Relative Motion and Meeting Point

Solving Equations Involving Definite Integrals