$f_{listener}=f_{source}dfrac{v_{sound}pm v_{listener}}{v_{sound}pm v_{source}}$

The rules are:

If the listener is travelling towards the source, the sign of $v_{listener}$ is positive. If it’s moving away, then it is negative.

Furthermore, if the source is moving towards from the listener, the sign of $v_{source}$ is negative. If the source is moving away from the listener, it is positive.

Since $v_{sound}=343;m/s$

$f_{listener}=f_{source}dfrac{343m/s}{343m/s-v_{source}}$

Solving for $v_{source}$

$$
v_{source}=(343;m/s)(1-dfrac{655;Hz}{702;Hz})=23.0m/s
$$

$$
begin{equation}
f_{text{observer}} = frac{f_{text{source}} }{left(1 pm frac{v_{text{source}}}{v_{text{sound}}} right)}
end{equation}
$$

Where the plus sign holds when the source is moving away from the observer,
and the minus sign when the source is moving toward the observer

The frequency of the sound the observer hears is $f_{text{observer}} = 590text{ Hz}$, and the speed of the emergency vehicle is $v = 23 ; frac{text{m}}{text{s}}$, so we use formula (1) to find the frequency produced by the siren $f_{text{source}}$.

We will use the formula with the plus sign, since the source is moving away from the observer.

$$
begin{align*}
f_{text{observer}} = frac{f_{text{source}} }{left(1 + frac{v_{text{source}}}{v_{text{sound}} } right)}
end{align*}
$$

Rearranging for $f_{text{source}}$ :

$$
begin{align*}
f_{text{source}} = f_{text{observer}} cdot left(1 + frac{v_{text{source}}}{v_{text{sound}}} right)
end{align*}
$$

Plugging in the values we get:

$$
begin{align*}
& f_{text{source}} = 590text{ Hz} cdot left(1 + frac{23 ; frac{text{m}}{text{s}}}{343 ; frac{text{m}}{text{s}} } right) \
& f_{text{source}} approx 630text{ Hz}
end{align*}
$$

$f_{listener}=f_{source}dfrac{v_{sound}pm v_{listener}}{v_{sound}pm v_{source}}$

The rules are:

If the listener is travelling towards the source, the sign of $v_{listener}$ is positive. If it’s moving away, then it is negative.

Furthermore, if the source is moving towards from the listener, the sign of $v_{source}$ is negative. If the source is moving away from the listener, it is positive.

$470=450left(dfrac{343}{343-v_{source}}right)$

Solve for $v_{source}$

$displaystyle v_{source} = 14.6 mathrm{m/s}$