Problem 73 - Prealgebra

Question

Digital Video The most popular video download of all time was a Star Wars movie trailer. The video was compressed so it would be small enough for people to download over the Internet. A formula for estimating the size, in kilobytes, of a compressed video is $$S=\frac{\text {height} \cdot \text {width} \cdot \text { fps } \cdot \text { time }}{35,000}$$ where height and width are in pixels, \(f p s\) is the number of frames per second the video is to play (television plays at 30 fps), and time is given in seconds. a. Estimate the size in kilobytes of the Star Wars trailer that has a height of 480 pixels, has a width of 216 pixels, plays at 30 fps, and runs for 150 seconds. b. Estimate the size in kilobytes of the Star Wars trailer that has a height of 320 pixels, has a width of 144 pixels, plays at 15 fps, and runs for 150 seconds.

Step-by-Step Solution

Answer

a) 1,283.66 KB; b) 296.23 KB.

Step 1: Understand the Formula

The formula for estimating the size of a compressed video in kilobytes is given by \( S = \frac{\text{height} \cdot \text{width} \cdot \text{fps} \cdot \text{time}}{35,000} \). This formula takes into account the dimensions of the video, the frames per second, and the duration in seconds to estimate the file size.

Step 2: Plug in Values for Part (a)

For the Star Wars trailer in part (a), insert the given values into the formula: height = 480 pixels, width = 216 pixels, fps = 30, and time = 150 seconds. The formula becomes \( S = \frac{480 \cdot 216 \cdot 30 \cdot 150}{35,000} \).

Step 3: Calculate the Product in the Numerator

First, calculate the product of all the values in the numerator: \( 480 \cdot 216 \cdot 30 \cdot 150 = 44,928,000 \).

Step 4: Divide to Find Part (a) Size

Divide the result from the numerator by 35,000 to find the size in kilobytes: \( S = \frac{44,928,000}{35,000} = 1,283.66 \) kilobytes.

Step 5: Plug in Values for Part (b)

Now, use the values for part (b): height = 320 pixels, width = 144 pixels, fps = 15, and time = 150 seconds. Substitute into the formula: \( S = \frac{320 \cdot 144 \cdot 15 \cdot 150}{35,000} \).

Step 6: Calculate the Product in the Numerator for Part (b)

Compute the product in the numerator for these values: \( 320 \cdot 144 \cdot 15 \cdot 150 = 10,368,000 \).

Step 7: Divide to Find Part (b) Size

Divide the result by 35,000 for the size: \( S = \frac{10,368,000}{35,000} = 296.23 \) kilobytes.

Key Concepts

Understanding Kilobytes

When you're working with video compression, understanding the unit "kilobytes" is essential. A kilobyte (KB) is a unit of digital information storage, which is often used to quantify the size of files or the amount of data being transmitted. In technical terms, one kilobyte is equivalent to 1,024 bytes, although in common usage it is often rounded to 1,000 bytes for simplicity.
For videos, determining the file size in kilobytes helps you gauge how much storage space it will take up, as well as its transfer speed over the internet. Thus, smaller file sizes generally mean faster download speeds and less storage required.
When applying our formula for the size of a video, kilobytes allow us to measure and understand the trade-offs between video quality and file size, which are critical in video compression.

Frames Per Second (fps)

Frames per second, or fps, is a crucial term in video compression that refers to how many individual images are displayed in one second of video. The higher the fps, the smoother the motion in the video will appear. Common standard television videos are typically at 30 fps, while some cinematic productions may go higher to achieve crisp visuals.
The fps not only affects the visual quality of the video but also its file size. Higher fps means more frames are stored in every second of the video, which can significantly increase the file size. For instance, a video at 30 fps will contain twice as many frames as one at 15 fps, assuming all other factors are equal.
When using the size estimation formula, fps is a direct multiplier in the calculation of the numerator, impacting the overall kilobyte size of the video file. Therefore, selecting the appropriate fps setting is an essential part of video production and compression decisions, balancing quality and size.

Formula Application in Video Compression

The video size estimation formula is a handy tool in understanding video compression, allowing us to predict file sizes given certain parameters: video width, height, fps, and duration. It's expressed as: \[ S = \frac{\text{height} \cdot \text{width} \cdot \text{fps} \cdot \text{time}}{35,000} \]This formula highlights how each of these factors influences the video file size. Let's break it down:

• Height and Width: In pixels, they define the resolution of the video. Larger resolutions translate into larger file sizes because more data is needed to capture every detail.
• Frames Per Second (fps): As discussed, more frames per second mean more images to store, increasing the file size.
• Time: This is the length of the video in seconds. Longer videos will naturally have larger file sizes due to more frames over time.

Using this formula helps video producers and editors plan out how large their video files will be before they’re created, especially when planning for uploads or downloads over bandwidth-limited networks. By inputting the components into the formula, we streamline estimating, which provides both efficiency and precision in managing digital video data.