Question

PLEASE SHOW ALL STEPS CLEARLY

1)To be able to fit 133 minutes’ worth of video on a single-sided single-layer DVD, a fair amount of compression is required. Calculate the compression factor required. Assume that 3.5 GB of space is available for the video track, that the image resolution is 720 × 480 pixels with 24-bit color (RGB at 8 bits each), and images are displayed at 30 frames/sec.

2)In a certain font, a monochrome laser printer can print 50 lines of 80 characters per page. The average character occupies a box 2 mm × 2 mm, about 25% of which is toner. The rest is blank (i.e., no toner) The toner layer is 25 microns thick. The printer’s toner cartridge measures 25 × 8 × 2 cm. How many pages is one toner cartridge good for?

Answer 1

1) Total no of pixels in one frame (n) = 720 * 480 = 345600

Every pixel requires 24 bits of memory (3 channels RGB and each channel of 8 bits)

∴ Total Memory required per frame = No of pixels in one frame * Memory required per pixel

= 345600 * 24 bits = 8294400 bits = 1036800 bytes (as 8 bits = 1 byte) = 1012.5 KB (as1 KB = 1024 bytes)

So, every frame requires 1012.5 KB of memory.

Total running time = 133 minutes = 133 * 60 seconds = 7980 seconds

Frame rate = 30 frames/second

∴ Total no of frames in 133 mins = Total running time * Frame rate = 7980 * 30 frames = 239400 frames

Now, 1012.5 KB of memory is required for each frame and there are 239400 frames for a video of 133 minutes.

∴ Total memory required = No of frames * Memory required per frame = 239400 * 1012.5 KB = 242392500 KB = 236711.4258 MB (As 1 MB = 1024 KB) = 231.1635 GB (As 1 GB = 1024 MB)

Thus a total memory of 231.1635 GB is required for 133 minutes of a video of resolution 720x480 pixels with 24-bit color, running at 30 frames/second.

But, available memory = 3.5 GB (As per the given data)

So, compression ratio = Memory required/Available memory = 231.1635/3.5 = 66.047 ≈ 66

So, the video has to be compressed by a factor of nearly 66 for it to fit on the DVD.

2) Number of characters per line C = 80
Number of lines per page L = 50
Therefore the number of characters per page (N) is given by N = L*C (1)
Each character occupies an area of 2mm (2 x 10^-3 m) by 2mm (2 x 10^-3 m) of which 25% is toner
So we can say the area used of toner to form each character (A) = 0.25 x (2 x 10^-3)*2 = 10^-6 m²
Toner layer is h = 25 microns (2.5 x 10^-5 m) thick
Volume of toner used for each character (Vc) = A*h = 2.5 x 10^-5 x 10^-6 m³ = 2.5 x 10^-11 m³
We know that (Vp) = N*Vc
Now on to the printer toner cartridge, this measures 25 x 8 x 2 cm or 0.25 x 0.08 x 0.02 m
Thus we can determine the volume of toner in 1 cartridge (V) = 0.25 x 0.08 x 0.02 = 4 x 10^-4 m³
Number of pages (Np) that one toner cartridge can supply Np = V/Vp = V/N*Vc
Substituting the value of N we get: Np = V/{LCVc} = 4 x 10^-4 / {50 x 80 x 2.5 x 10^-11 }
Np = 4000 pages per toner cartridge

Hope this helps.