Question

A movie with an aspect ratio of 1.85:1 is shown as a letterboxed image on a newer 50-inch 16:9 television. Calculate the height of the image, the height of each barely visible black bar at the top and bottom of the screen, and the percent of the screen’s area that is occupied by the image. Use a variety of representations to justify your response.

Answer 1

The aspect ratio of a Television is the ratio of the width to the height of the television.

• The height of image on the 50-inch television is 24.48 inches
• The height of each black bar is 23.52 inches
• The percentage of the screen area occupied is 96.08%

The size of a TV is calculated using Pythagoras theorem. Assume the length ratio of the new 50-inch 16:9 TV is x.

The ratio is represented as:

$Width : Height = 16 : 9$

Using Pythagoras theorem, we have:

$(16x)^2 + (9x)^2 = 50^2$

$256x^2 + 81x^2 = 2500$

$337x^2 = 2500$

Divide both sides by 337

$x^2 = 7.42$

Take square roots of both sides

$x = \sqrt{7.42$

$x = 2.72$

So, the width of the image is:

$Width = 16x$

$Width = 16 \times 2.72$

$Width = 43.52$

The height of the image is then calculated as:

$Height = 9x$

$Height = 9 \times 2.72$

$Height = 24.48$

The height of the image is 24.48 inches

The height of each black bar is calculated as follows:

$Width : Height = 1.85 : 1$

Express as fraction

$\frac{Height}{Width}= \frac{1}{1.85}$

Make Height the subject

$Height = \frac{1}{1.85}\times Width$

Substitute $Width = 43.52$

$Height = \frac{1}{1.85}\times 43.52$

$Height = 23.52$

The height of each black bar is 23.52 inches

Lastly, the percentage of the screen’s area that is occupied by the image

First, we calculate the area of the bars

$A_1 = Height \times Width$

Where:

$Height = 23.52$ and $Width = 43.52$

So:

$A_1 = 23.52 \times 43.52$

$A_1 = 1023.59$

Next, calculate the area of the 50-inch TV

$A_2 = Height \times Width$

Where

$Width = 43.52$ and $Height = 24.48$

So:

$A_2 = 24.48 \times 43.52$

$A_2 = 1065.37$

So, the percentage occupied on the screen area is:

$\% Screen = \frac{A_1}{A_2} \times 100\%$

$\% Screen = \frac{1023.59}{1065.37} \times 100\%$

$\% Screen = \frac{102359}{1065.37} \%$

$\% Screen = 96.08 \%$

Hence, the percentage of the screen area occupied is 96.08%

Read more about aspect ratios at:

https://brainly.com/question/17231027