An operator seated in his base camp transmits a sound signal of...

Let the speed of the car be $v_c$.

The car acts as a moving observer receiving the sound from the stationary base camp. The frequency received by the car is $f' = f \left( \dfrac{v + v_c}{v} \right).<br><br>The car then reflects the sound, acting as a moving source. The frequency received by the stationary base camp is$f'' = f' \left( \dfrac{v}{v - v_c} \right)$.<br><br>Combining these, the apparent frequency of the reflected sound is$f'' = f \left( \dfrac{v + v_c}{v - v_c} \right).

Substituting the given values $f = 400$\text{ Hz}, $f'' = 410$\text{ Hz}, and $v = 324$\text{ m s}^{-1}:

$410 = 400 \left( \dfrac{324 + v_c}{324 - v_c} \right)$

$\dfrac{41}{40} = \dfrac{324 + v_c}{324 - v_c}$

$41(324 - v_c) = 40(324 + v_c)$

$41 \times 324 - 41 v_c = 40 \times 324 + 40 v_c$

$324(41 - 40) = 81 v_c$

$324 = 81 v_c$

$v_c = 4 \text{ m s}^{-1}$