Chapter SH, Problem 13.8APP

To determine

To calculate: Ratio of the reflecting area of the mirror with a hole to the reflecting area of the same mirror without the hole.

Answer to Problem 13.8APP

Ratio of the reflecting area of the mirror with a hole to the reflecting area of the same mirror without the hole is $\underset{\_}{24:25}$ .

Explanation of Solution

Given:

Diameter of concave mirror = $50\text{ cm}$

Diameter of hole = $10\text{ cm}$

Formula used:

Area of a circle is given by,

  $\text{Area}=\pi r^2$

Where r is radius.

Calculation:

For the concave mirror,

  $\text{Radius = }\frac{\text{Diameter}}{2}=\frac{\text{50 cm}}{2}=25\text{ cm}$

Area is calculated as:

  $\begin{array}{l}\text{ Area}=\pi r^2\\ =\pi {\left(25\text{ cm}\right)}^2\\ =625\pi {\text{ cm}}^2\end{array}$

For the hole,

  $\text{Radius = }\frac{\text{Diameter}}{2}=\frac{\text{10 cm}}{2}=5\text{ cm}$

Area is calculated as:

  $\begin{array}{l}\text{ Area}=\pi r^2\\ =\pi {\left(\text{5 cm}\right)}^2\\ =25\pi {\text{ cm}}^2\end{array}$

Area of reflecting surface of the concave mirror with the hole is,

  $\begin{array}{l}\text{ Area of concave mirror}-\text{Area of hole}\\ =625\pi -25\pi \\ =600\pi {\text{ cm}}^2\end{array}$

Ratio of the reflecting area of the mirror with a hole to the reflecting area of the same mirror without the hole is calculated as:

  $\begin{array}{l}\frac{\text{Area of concave mirror with hole}}{\text{Area of concave mirror without hole}}\\ =\frac{600\pi {\text{ cm}}^2}{625\pi {\text{ cm}}^2}\\ =\frac{24}{25}\end{array}$

Conclusion:

Ratio of the reflecting area of the mirror with a hole to the reflecting area of the same mirror without the hole is $\underset{\_}{24:25}$ .