Answer
$$\Delta t = L(\frac{v_m-v}{v_mv})$$
Work Step by Step
An equation relating distance, time, and velocity is the velocity equation, which is $$v=\frac{\Delta x}{\Delta t}$$ Solving for $\Delta t$ yields $$\Delta t=\frac{\Delta x}{v}$$ Finding the time to travel through each substance is found using this equation. $$\Delta t_{air}=\frac{L}{v}$$ $$\Delta t_{metal}=\frac{L}{v_m}$$ Since the speed of sound in a metal is greater than the speed of sound in air, subtract the time through metal from the time in air. This leaves $$\Delta t=\frac{L}{v}-\frac{L}{v_m}$$ Finding a common denominator and combining the fractions yields $$\Delta t=\frac{Lv_m}{v_mv}-\frac{Lv}{v_mv}=L(\frac{v_m-v}{v_mv})$$
