Chapter 9.2, Problem 61PPS

a.

To determine

Tofind: the rational expression that representing the difference in pitch between the two sirens.

Answer to Problem 61PPS

The rational expression is $\frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}$ that representing the difference in pitch between the two sirens.

Explanation of Solution

Given information:

  $x$ represents the speed of the faster fire engine and $y$ represents the speed of the slower fire engine.

Calculation: since, $x$ represents the speed of the faster fire engine and $y$ represents the speed of the slower fire engine.

Doppler effect of the faster fire engine is represented by the ration expression $=\frac{P_0{S}_0}{\left(S_0-x\right)}$

Doppler effect of the slower fire engine is represented by the rational expression $=\frac{P_0{S}_0}{\left(S_0-y\right)}$

Now, subtract both expressions,

  $\begin{array}{l}\frac{P_0{S}_0}{\left(S_0-x\right)}-\frac{P_0{S}_0}{\left(S_0-y\right)}=\frac{P_0{S}_0\left(S_0-y\right)-P_0{S}_0\left(S_0-x\right)}{\left(S_0-x\right)\left(S_0-y\right)}\text{ [LCM]}\\ \frac{P_0{S}_0}{\left(S_0-x\right)}-\frac{P_0{S}_0}{\left(S_0-y\right)}=\frac{P_0{S}_0\left[S_0-y-S_0+x\right]}{\left(S_0-x\right)\left(S_0-y\right)}\text{ [simplify]}\\ \frac{P_0{S}_0}{\left(S_0-x\right)}-\frac{P_0{S}_0}{\left(S_0-y\right)}=\frac{P_0{S}_0\left(x-y\right)}{\left(S_0-x\right)\left(S_0-y\right)}\text{ [subtract]}\\ \frac{P_0{S}_0}{\left(S_0-x\right)}-\frac{P_0{S}_0}{\left(S_0-y\right)}=\frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}\text{ [multiply]}\end{array}$

Thus, the expression is $\frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}$ .

b.

To determine

To find: the difference between the pitches if one is travelling at $45$ meters per second and other is travelling at $70$ meters per second.

Answer to Problem 61PPS

The difference between the pitches is $55.2$ if one is travelling at $45$ meters per second and other is travelling at $70$ meters per second.

Explanation of Solution

Given information:

The rational expression is $\frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}$ that representing the difference in pitch between the two sirens.

Calculation: to find the difference first substitute $x=70,y=45,S_0=332,P_0=500$ and simplify,

  $\begin{array}{l}\frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}=\frac{\left(500\right)\left(332\right)\left(70\right)-\left(500\right)\left(332\right)\left(45\right)}{\left(332-70\right)\left(332-45\right)}\text{ [substitute]}\\ \frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}=\frac{11620000-7470000}{\left(262\right)\left(287\right)}\text{ [multiply]}\\ \frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}=\frac{4150000}{75194}\text{ [subtract]}\\ \frac{P_0{S}_{0}x-P_0{S}_{0}y}{\left(S_0-x\right)\left(S_0-y\right)}=55.2\text{ [divide]}\end{array}$

Thus, the difference between pitches is $55.2$ .