Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.


b. When, if ever, does Alicia overtake Boris?

Verified step by step guidance

1. First, understand that to find when Alicia overtakes Boris, we need to find the time \( t \) when both have run the same distance from the starting point of Alicia. Since Boris has a 2-mile head start, his position at time \( t \) will be his initial 2 miles plus the distance he covers after time \( t \).
2. Express the position functions for both runners by integrating their velocity functions over time. For Alicia, her position function \( s_A(t) \) is given by integrating \( v(t) = \frac{4}{t+1} \) from 0 to \( t \): \[ s_A(t) = \int_0^t \frac{4}{x+1} \, dx \]
3. Similarly, for Boris, his position function \( s_B(t) \) includes his 2-mile head start plus the integral of his velocity function \( u(t) = \frac{2}{t+1} \) from 0 to \( t \): \[ s_B(t) = 2 + \int_0^t \frac{2}{x+1} \, dx \]
4. Calculate both integrals to find explicit expressions for \( s_A(t) \) and \( s_B(t) \). These will involve natural logarithms because the integral of \( \frac{1}{x+1} \) is \( \ln|x+1| \).
5. Set the two position functions equal to each other to find the time when Alicia overtakes Boris: \[ s_A(t) = s_B(t) \] Solve this equation for \( t \) to find the time(s) when Alicia catches up to Boris.

Key Concepts

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. To find the position function from a velocity function, integrate the velocity over time. This allows us to determine the location of each runner at any time t, which is essential for comparing their positions.

Solving Equations to Find Intersection Points

To find when Alicia overtakes Boris, we set their position functions equal and solve for t. This involves solving an equation that may include algebraic manipulation and understanding the domain of the functions. The solution gives the time(s) when both runners are at the same position.

Handling Velocity Functions with Variable Denominators

The given velocity functions have the form v(t) = constant / (t + 1), which affects how we integrate them. Recognizing this form helps in applying the correct integration techniques, such as substitution, to find the position functions accurately.