91Ó°ÊÓ

The path length the person took denoted by x is in the range of 0 ≤x≤150 cm.

The sound is reinforced to its maximum when the path difference between the sound waves is an integral multiple of the wavelength. This is also known as the Constructive interference. The condition is,

${\mathrm{r}}_2-{\mathrm{r}}_1=\mathrm{³¾Î»},      \mathrm{m}=0,Â\pm 1,Â\pm 2,...$ ...(i)

Here, in this problem ${\mathrm{r}}_2$= 150 cm and ${\mathrm{r}}_2$= x cm.

Substitute the values in equation (i),

$$\begin{gathered} 150\mathrm{cm}-\mathrm{x}=\mathrm{m}(34\mathrm{cm}) \\ \mathrm{x}=150\mathrm{cm}-\mathrm{m}(34\mathrm{cm}) \end{gathered}$$

Thus, the values of x are,

localid="1663912422978" $$\begin{gathered} form=0 \\ \mathrm{x}=150\mathrm{cm} \end{gathered}$$

$$\begin{gathered} form=1 \\ x=150\mathrm{cm}-34\mathrm{cm} \\ =116\mathrm{cm} \end{gathered}$$

localid="1663912531079" $$\begin{gathered} \mathrm{for}\mathrm{m}=2 \\ \mathrm{x}=150\mathrm{cm}-2\left(34\right)\mathrm{cm} \\ =82\mathrm{cm} \end{gathered}$$

$$\begin{gathered} form=3 \\ x=150\mathrm{cm}-3\left(34\right)\mathrm{cm} \\ =48\mathrm{cm} \end{gathered}$$

$$\begin{gathered} form=4 \\ x=150\mathrm{cm}-4\left(34\right)\mathrm{cm} \\ =14\mathrm{cm} \end{gathered}$$

The sound is cancelled when the path difference between the sound waves is a multiple of the odd number of half wavelengths. This is also known as the destructive interference. The condition is,

${\mathrm{r}}_2-{\mathrm{r}}_1=\left(\mathrm{m}+\frac{1}{2}\right)\mathrm{λ},      \mathrm{m}=0,Â\pm 1,Â\pm 2,...$ ...(ii)

Here, in this problem${\mathrm{r}}_2$= 150 cm and${\mathrm{r}}_1$= x cm.

Substitute the values in equation (ii),

$$\begin{gathered} 150\mathrm{cm}-\mathrm{x}=\left(\mathrm{m}+\frac{1}{2}\right)(34\mathrm{cm}) \\ \mathrm{x}=150\mathrm{cm}-\left(\mathrm{m}+\frac{1}{2}\right)(34 \mathrm{cm}) \end{gathered}$$

$$\begin{gathered} form=0 \\ \mathrm{x}=133\mathrm{cm} \\ \mathrm{for}\mathrm{m}=1 \\ \mathrm{x}=150\mathrm{cm}-\frac{3}{2}\left(34\right)\mathrm{cm} \\ =99\mathrm{cm} \\ \mathrm{for}\mathrm{m}=2 \\ \mathrm{x}=150\mathrm{cm}-\frac{5}{2}\left(34\right)\mathrm{cm} \\ =65\mathrm{cm} \\ \mathrm{for}\mathrm{m}=3 \\ \mathrm{x}=150\mathrm{cm}-\frac{7}{2}\left(34\right)\mathrm{cm} \\ =31\mathrm{cm} \end{gathered}$$

Thus, The values of x for which the sound will be cancelled is 133 cm, 99 cm, 65 cm, 31 cm.