Chapter 23, Problem 44P

To determine

Distance from Hannah to images of four candles in her view.

Answer to Problem 44P

Distance from Hannah to images of four candles in her view will be $36.5\text{m}$.

Explanation of Solution

First image is the result of a single reflection. Write the equation to find the distance of first image from Hannah.

  $d_1=d_{wall}-d_{can}+d_{Hanna}$                                                                                              (I)

Here, $d_1$ is the distance of first image from Hannah, $d_{wall}$ is the separation between two mirrors, $d_{can}$ is the distance from candle to nearer mirror, and $d_{Hanna}$ is the distance from Hannah to each mirror.

Second image is the result of two reflections. Write the equation to find the distance of second image from Hannah.

    $d_2=d_{wall}+d_{can}+d_{Hanna}$                                                                              (II)

Here, $d_2$ is the distance of second image from Hannah.

Third image is the result of three reflections. Write the equation to find the distance of third image from Hannah.

    $d_3=d_{wall}+d_{wall}+\left(d_{wall}-d_{can}\right)+d_{Hanna}$                                                                   (III)

Here, $d_3$ is the distance of third image from Hannah.

Fourth image is the result of four reflections. Write the equation to find the distance of fourth image from Hannah.

    $d_4=d_{wall}+d_{wall}+d_{wall}+d_{can}+d_{Hanna}$                                                                    (IV)

Here, $d_4$ is the distance of fourth image from Hannah.

Conclusion:

Substitute $10.0\text{m}$ for $d_{wall}$, $1.5\text{m}$ for $d_{can}$, and $5.0\text{m}$ for $d_{can}$ in equation (I) to find $d_1$.

  $\begin{array}{c}{d}_1=10.0\text{m}-1.5\text{m}+5.0\text{m}\\ =13.5\text{m}\end{array}$

Substitute $10.0\text{m}$ for $d_{wall}$, $1.5\text{m}$ for $d_{can}$, and $5.0\text{m}$ for $d_{can}$ in equation (II) to find $d_2$.

  $\begin{array}{c}{d}_2=10.0\text{m+}1.5\text{m}+5.0\text{m}\\ =16.5\text{m}\end{array}$

Substitute $10.0\text{m}$ for $d_{wall}$, $1.5\text{m}$ for $d_{can}$, and $5.0\text{m}$ for $d_{can}$ in equation (III) to find $d_3$.

  $\begin{array}{c}{d}_3=10.0\text{m}+10.0\text{m}+\left(10.0\text{m}-1.5\text{m}\right)+5.0\text{m}\\ \text{=33}\text{.5}\text{m}\end{array}$

Substitute $10.0\text{m}$ for $d_{wall}$, $1.5\text{m}$ for $d_{can}$, and $5.0\text{m}$ for $d_{can}$ in equation (IV) to find $d_4$.

  $\begin{array}{c}{d}_4=10.0\text{m}+10.0\text{m}+10.0\text{m}+5.0\text{m}+1.5\text{m}\\ \text{=36}\text{.5}\text{m}\end{array}$

Therefore, distance from Hannah to images of four candles in her view will be $36.5\text{m}$.