Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Frames in a movie film 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 movie has approximately \(4.1472 \times 10^6\) frames.

Step 1: Identify the Movie Duration in Seconds

The movie runs for 48 hours. First, we need to convert this time into seconds since the frame rate is measured per second. We know that 1 hour has 60 minutes, and each minute has 60 seconds. Therefore, 48 hours is:\[48 \text{ hours} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 172,800 \text{ seconds}\]

Step 2: Determine the Total Number of Frames

With the film speed given as 24 frames per second, we multiply the total duration of the film in seconds by the number of frames per second to find the total number of frames:\[172,800 \text{ seconds} \times 24 \text{ frames/second} = 4,147,200 \text{ frames}\]

Step 3: Express in Scientific Notation

To express the total number of frames in scientific notation, identify the significant digits and the appropriate power of 10:The number 4,147,200 can be written in scientific form as:\[4.1472 \times 10^6 \text{ frames}\]

Key Concepts

Frame Rate Calculations

Frame rate is the number of frames displayed per second of film or video. Understanding frame rate calculations can be crucial in a variety of fields, be it videography, animations, or mathematics itself. To calculate the total number of frames in a film, we multiply the film's duration in seconds by the frame rate.
For example, if a movie runs at 24 frames per second and lasts for 48 hours, we first need to convert the hours into seconds and then calculate the total frames.
Frame rate calculations help in understanding how video playback correlates with real-time, offering insights into digital media production. It ensures consistency in video quality and can also help in determining the storage requirements since the total frames affect the file size.

Time Conversion

Time conversion is a fundamental skill applied to many real-world scenarios where we need to translate one time unit into another. In this exercise, we need to convert hours into seconds to match the unit of measurement for frame rate.
Remember that:

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

So, when converting 48 hours to seconds, you calculate it by multiplying:
1. 48 hours by 60 to get the minutes: 48 x 60 minutes
2. Then multiply the result by 60 to convert the minutes to seconds.
Using conversions correctly ensures that calculations are consistent and scalable, making it easier to work with complex time-based problems.

Significant Figures

Significant figures are important in scientific calculations because they reflect the precision of the measurement. When expressing numbers in scientific notation as asked in this problem, you must consider significant figures carefully.
Let's explore how to express a large number like 4,147,200 into scientific notation while observing significant figures:

• Identify non-zero digits (in this case, 41472).
• Place a decimal after the first non-zero digit: 4.1472.
• Count the total amount of digits after the first non-zero digit to use as the power of ten.

Thus, 4,147,200 converts to scientific notation as \(4.1472 \times 10^6\), maintaining significant digits.
Understanding significant figures prevents inaccuracies and ensures that you are representing numbers with the correct level of precision, which is invaluable in both scientific and practical calculations.