Chapter 31, Problem 75P

(a)

To determine

The value of $x$ .

Answer to Problem 75P

The value of $x$ is $-1.00\text{m}$ .

Explanation of Solution

Formula used:

The expression for distance $D$ is given by,

  $D=d_1+d_2$

The expression for distance $d_1$ is given by,

  $d_1=\sqrt{{\left(x+2\right)}^2+4}$

The expression for distance $d_2$ is given by,

  $d_2=\sqrt{{\left(2-x\right)}^2+36}$

Calculation:

The distance $D$ is calculated as,

  $\begin{array}{c}D=d_1+d_2\\ =\sqrt{{\left(x+2\right)}^2+4}+\sqrt{{\left(2-x\right)}^2+36}\end{array}$ ............ (1)

Differentiate equation (1) with respect to $x$ .

  $\begin{array}{c}\frac{dD}{dx}=\frac{d}{dx}\left[\sqrt{{\left(x+2\right)}^2+4}+\sqrt{{\left(2-x\right)}^2+36}\right]\\ =\frac{1}{2}{\left[{\left(x+2\right)}^2+4\right]}^{-\frac{1}{2}}2\left(x+2\right)+\frac{1}{2}{\left[{\left(2-x\right)}^2+36\right]}^{-\frac{1}{2}}2\left(2-x\right)\left(-1\right)\end{array}$ ............ (2)

Substitute $0$ for $\frac{dD}{dx}$ in equation (2).

  $\begin{array}{c}\frac{1}{2}{\left[{\left(x+2\right)}^2+4\right]}^{-\frac{1}{2}}2\left(x+2\right)+\frac{1}{2}{\left[{\left(2-x\right)}^2+36\right]}^{-\frac{1}{2}}2\left(2-x\right)\left(-1\right)=0\\ \frac{x+2}{\sqrt{{\left(x+2\right)}^2+4}}-\frac{2-x}{\sqrt{{\left(2-x\right)}^2+36}}=0\\ x=-1.00\text{m}\end{array}$

Conclusion:

Therefore, the value of $x$ is $-1.00\text{m}$ .

(b)

To determine

The angle of incidence.

Answer to Problem 75P

The angle of incidence is $26.6°$ .

Explanation of Solution

Formula used:

The expression for angle for incidence is given by,

  ${\theta }_{\text{i}}={\tan }^{-1}\left(\left|\frac{x-x_1}{y-y_1}\right|\right)$

Calculation:

The angle of incidence is calculated as,

  $\begin{array}{c}{\theta }_{\text{i}}={\tan }^{-1}\left(\left|\frac{x-x_1}{y-y_1}\right|\right)\\ ={\tan }^{-1}\left(\left|\frac{-1.00\text{m}-\left(-2.00\text{m}\right)}{0-2.00\text{m}}\right|\right)\\ =26.6°\end{array}$

Conclusion:

Therefore, the angle of incidence is $26.6°$ .

(c)

To determine

The angle of reflection.

Answer to Problem 75P

The angle of reflection is $26.6°$ .

Explanation of Solution

Formula used:

The expression for angle for reflection is given by,

  ${\theta }_{\text{i}}={\theta }_{\text{r}}$

Calculation:

The angle of reflection is calculated as,

  $\begin{array}{c}{\theta }_{\text{i}}={\theta }_{\text{r}}\\ =26.6°\end{array}$

Conclusion:

Therefore, the angle of reflection is $26.6°$ .