Chapter 23, Problem 55P

(a)

To determine

The radius of curvature of the mirror.

Answer to Problem 55P

The radius of curvature of the mirror is $3.9\text{m}$.

Explanation of Solution

Write an expression for the magnification.

  $\frac{h\text{'}}{h}=-\frac{q}{p}$                 (I)

Here, $h\text{'}$ is the image height, $h$ is the object height, $q$ is the image distance and $p$ is the object distance.

Rearrange the equation (I) to find image distance.

    $q=-p\frac{h\text{'}}{h}$      (II)

Write an expression for focal length.

    $f=-\frac{1}{2}R$                                                                                                               (III)

Here, $f$ is the focal length and $R$ is the radius of curvature of the mirror.

Write an expression for mirror equation.

    $\frac{1}{f}=\frac{1}{p}+\frac{1}{q}$                                                                                                             (IV)

Substitute equation (II) and (III) in equation (IV).

    $\frac{1}{\left(-\frac{1}{2}R\right)}=\frac{1}{p}+\frac{1}{\left(-p\frac{h\text{'}}{h}\right)}$                                                                                         (V)

Rearrange the equation (V) to find the radius of curvature.

    $R=\frac{2}{\frac{h}{ph\text{'}}-\frac{1}{p}}$                                                                                                         (VI)

Conclusion:

Substitute $1.64\text{m}$ for $h$, $4.5\text{m}$ for $p$ and $0.500\text{m}$ for $h\text{'}$ in equation (VI) to find $R$.

    $\begin{array}{c}R=\frac{2}{\frac{\left(1.64\text{m}\right)}{\left(4.5\text{m}\right)\left(0.500\text{m}\right)}-\frac{1}{\left(4.5\text{m}\right)}}\\ =\frac{2\left(4.5\text{m}\right)\left(0.500\text{m}\right)\left(4.5\text{m}\right)}{\left(1.64\text{m}\right)\left(4.5\text{m}\right)-\left(4.5\text{m}\right)\left(0.500\text{m}\right)}\\ =3.9\text{m}\end{array}$

Thus, the radius of curvature of the mirror is $3.9\text{m}$.

(b)

To determine

If the image of the woman fill the mirror from top to bottom.

Answer to Problem 55P

The image of the mirror does not fill the mirror.

Explanation of Solution

Write an expression for the image distance.

    $q=-p\frac{h\text{'}}{h}$                                                                                                         (VII)

Conclusion:

Substitute $1.64\text{m}$ for $h$, $4.5\text{m}$ for $p$ and $0.500\text{m}$ for $h\text{'}$ in equation (VII) to find $q$.

    $\begin{array}{c}R=\frac{\left(4.5\text{m}\right)\left(0.500\text{m}\right)}{\left(1.64\text{m}\right)}\\ =-\frac{2.25{\text{m}}^{\text{2}}}{1.64\text{m}}\\ =-1.4\text{m}\end{array}$

Thus, the image appears $1.4\text{m}$ behind the mirror. The diameter of face of mirror and the height of mirror should be equal in order to fit the image completely in the mirror. However, the image height is less since the image is formed $1.4\text{m}$ behind the mirror. Thus, the image of the mirror does not fill the mirror.