A system with a 9" input screen and a 1" output screen increases brightness by how many times?

The brightness increase of a system with a given input and output screen size can be determined by the ratio of the areas of the two screens. The area of a circle is calculated using the formula ( A = \pi r^2 ), where ( r ) is the radius.

For the input screen of 9 inches in diameter, the radius is 4.5 inches. Thus, the area of the input screen can be calculated as:

[

A_{\text{input}} = \pi (4.5)^2 = \pi \times 20.25

]

For the output screen of 1 inch in diameter, the radius is 0.5 inches. The area of this screen is:

[

A_{\text{output}} = \pi (0.5)^2 = \pi \times 0.25

]

Now, calculating the ratio of the areas gives:

[

\text{Brightness increase} = \frac{A_{\text{input}}}{A_{\text{output}}} = \frac{\pi \times 20.25}{\pi \times 0.25} = \frac{20.25}{0.25} = 81

\