Problem 59 - Precalculus: Functions and Graphs

Question

One of the longest movies ever made is a 1970 British film that runs for 48 hours. Assuming that the film speed is 24 frames per second, approximate the total number of frames in this film. Express your answer in scientific form.

Step-by-Step Solution

Answer

The film has approximately \(4.15 \times 10^6\) frames.

Step 1: Calculate Total Seconds

First, convert the movie duration from hours to total seconds. We know there are 60 minutes in an hour and 60 seconds in a minute. Therefore, in 48 hours: \[\text{Total seconds} = 48 \times 60 \times 60 \]

Step 2: Determine Total Frames

Next, calculate the total number of frames by multiplying the total number of seconds from Step 1 by the frame rate of 24 frames per second: \[\text{Total frames} = \text{Total seconds} \times 24 \]

Step 3: Convert to Scientific Notation

Finally, express the total frames in scientific notation. For example, if the total frames calculated is 4,147,200, then in scientific notation it is written as:\[4.1472 \times 10^6\]

Key Concepts

Frame Rate Calculation

Frame rate is a crucial element in film and video production as it determines the smoothness of the visual playback. It is quantified as the number of frames per second (fps), which tells us how many individual frames or images make up one second of video.
For instance, in the provided exercise, the film is played at a frame rate of 24 fps. This means that for every second, 24 distinct images are shown in succession to create the illusion of motion. The importance of understanding this concept lies in its ability to help predict how many frames are needed for a given length of time. To calculate the total frames within a specific time frame, you simply multiply the duration in seconds by the fps.
This is vital for film editors and producers who need to manage storage and processing requirements. Understanding frame rate also helps in deciding the aesthetic feel of the video, with different fps yielding different cinematic effects.

Time Conversion

Time conversion is a method used to change time measurements between various units, such as from hours to seconds. It is a fundamental skill in various applications, including physics, engineering, and everyday problem-solving.
In our exercise, we need to convert 48 hours of movie runtime into seconds to calculate the total frames.The conversion process involves using known ratios. For time, these conversions are simple:

• 1 hour equals 60 minutes.
• 1 minute equals 60 seconds.

To find the total seconds in 48 hours, multiply:\[48 \text{ hours} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute}\] This yields a total of 172,800 seconds.
Converting time before further calculations ensures accuracy and consistency in the result, especially when dealing with large data sets.

Multiplication in Scientific Notation

Scientific notation is a method of representing numbers that are too large or too small in a concise form. It is especially useful in scientific and engineering calculations. To express a number in scientific notation, it is written as a product of a decimal and a power of ten. In our example, after calculating the total frames as a large number (e.g., 4,147,200), we convert it to scientific notation for simplicity. Let's see how this is done:

• Identify the significant figures, which in this case are 4.1472.
• Count the number of places the decimal point moves to give: 4.1472 becomes 4.1472 × 10⁶.

Scientific notation makes it easier to handle calculations and comparisons without dealing with a long string of digits.
It's particularly advantageous when working with technologies that need precision and small storage space. Moreover, it provides a standardized way of communicating large numbers.