Chapter SH, Problem 13.4APP

To determine

To calculate: Ratio of the area of the Keck telescope mirror to the area of a dilated human pupil.

Answer to Problem 13.4APP

Ratio of the area of the Keck telescope mirror to the area of a dilated human pupil is $\underset{\_}{1000000:1}$ .

Explanation of Solution

Given:

Diameter of dilated pupil = $\text{1 cm}$

Diameter of Keck telescope mirror = $10\text{ m}$

Formula used:

Area of a circle is given by,

  $\text{Area}=\pi r^2$

Where r is radius.

Calculation:

For the dilated pupil,

  $\text{Radius = }\frac{\text{Diameter}}{2}=\frac{1\text{ cm}}{2}=0.5\text{ cm}$

Area is calculated as:

  $\begin{array}{l}\text{ Area}=\pi r^2\\ =\pi {\left(0.5\text{ cm}\right)}^2\\ =0.25\pi {\text{ cm}}^2\end{array}$

For the Keck telescope mirror,

  $\text{Radius = }\frac{\text{Diameter}}{2}=\frac{\text{10 m}}{2}=5\text{ m}=500\text{ cm}$

Area is calculated as:

  $\begin{array}{l}\text{ Area}=\pi r^2\\ =\pi {\left(\text{500 cm}\right)}^2\\ =250000\pi {\text{ cm}}^2\end{array}$

Ratio of the area of the Keck telescope mirror to the area of a dilated human pupil is calculated as:

  $\begin{array}{l}\frac{\text{Area of Keck telescope mirror}}{\text{Area of dilated pupil}}\\ =\frac{250000\pi {\text{ cm}}^2}{0.25\pi {\text{ cm}}^2}\\ =\frac{1,000,000}{1}\end{array}$

Conclusion:

Ratio of the area of the Keck telescope mirror to the area of a dilated human pupil is $\underset{\_}{1000000:1}$ .