## Step 1: Model the Problem Mathematically Let \( f(t) \) be your position at time \( t \), and \( g(t) \) be your friend's position at time \( t \), where both start at the same time (\( t = \)), at the same place (\( f() = g() = \)), and finish together (\( f(T) = g(T) = 10 \) km for some time \( T \)). Let \( h(t) = f(t) - g(t) \). So, \( h() = \) and \( h(T) = \). **Explanation:** We have defined a function \( h(t) \) measures the difference in your positions at any time \( t \). Since you start and finish together, this difference is zero at the beginning and the end. --- ## Step 2: Apply Rolle’s Theorem Since \( h(t) \) is continuous and differentiable (because both you and your friend are running and their positions are continuous functions of time), and \( h() = h(T) = \), **Rolle’s Theorem** applies. Therefore, there must be some time \( c \) in \( (, T) \) such that \( h'(c) = \). But \( h'(t) = f'(t) - g'(t) \), which means at time \( c \), \( f'(c) = g'(c) \). **Explanation:** Rolle's Theorem guarantees that, at some instant between starting and finishing, the rate of change of the position difference is zero—meaning both runners had the same speed at that instant. --- ## Final Answer **There must be at least one moment during the race where you and your friend were running at the same speed.**