Answer
The speed of the race car is $49.0~m/s$
Work Step by Step
Let $v_c$ be the race car's speed. We can use the equation for the Doppler effect when the observer is approaching to find an expression for the higher frequency $f_h$:
$f_h = \left(\frac{v+v_c}{v}\right)~f_s$
We can use the equation for the Doppler effect when the observer is moving away to find an expression for the lower frequency $f_l$:
$f_l = \left(\frac{v-v_c}{v}\right)~f_s$
Note that $f_l = 0.75~f_h$:
$f_l = 0.75~f_h$
$\left(\frac{v-v_c}{v}\right)~f_s = 0.75~\left(\frac{v+v_c}{v}\right)~f_s$
$v-v_c =0.75~(v+v_c)$
$1.75~v_c = 0.25~v$
$v_c = \frac{0.25~v}{1.75}$
$v_c = \frac{(0.25)~(343~m/s)}{1.75}$
$v_c = 49.0~m/s$
The speed of the race car is $49.0~m/s$.
