Question

In Exercises \(40-42,\) you are working as an animator. Each frame in an animated feature film takes at least one hour to draw. When projected, 35-millimeter film runs at 24 frames per second. A 2 \(\frac{1}{2}\) -hour movie has about \(216,000\) frames. Write an inequality that describes the number of hours it would take to draw the frames needed for a \(1 \frac{1}{2}\) -hour animated feature film.

Short answer

The number of hours it would take to draw the frames for a 1.5-hour feature film is at least 129600 hours, represented by the inequality \(H \geq 129600\).

Step-by-step solution

Convert film length to seconds

Convert the length of the film from hours to seconds. Since 1 hour = 3600 seconds, a 1.5 hour film is \(1.5 \times 3600 = 5400\) seconds.

Calculate number of frames

Calculate the total number of frames by multiplying the number of seconds by the frame rate (24 frames per second). Hence, the number of frames required is \(5400 \times 24 = 129600\).

Create inequality

As each frame takes at least an hour to draw, the number of hours needed to draw 129600 frames is at least 129600 hours. Therefore, if H represents the number of hours, the inequality is \(H \geq 129600\).

Key concepts

Frame Rate Conversion

Understanding frame rate conversion is key when dealing with multimedia content such as animated films or videos. In the context of animation, the frame rate refers to the number of individual images, or frames, that are displayed per second in order to create the illusion of motion.

Commonly, this rate is standardized for various mediums; for example, 24 frames per second (fps) is a standard frame rate for 35mm films. Converting frame rates between different mediums or determining how many frames are needed for a certain duration of time requires basic multiplication. If you have a movie that is 1.5 hours long and you know that the frame rate is 24 fps, you simply multiply the duration in seconds (after converting from hours to seconds) by the frame rate to find the total number of frames.

To improve understanding, remember to keep units consistent when converting frame rates. When given time in hours and a frame rate per second, it's crucial to convert hours to seconds before performing multiplication. This will ensure you're correctly calculating the total number of frames.

Inequality Representation

In algebra, an inequality is a statement that describes the relative size or order of two values. It shows if one number is greater than, less than, or possibly equal (in the case of 'at least' or 'no more than') to another. Inequalities are represented using symbols like <, >, \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to).

When translating real-world situations into algebraic inequalities, start by identifying what you are comparing. In our example, the task involves determining the minimum number of hours required to draw frames for an animated film, where the critical piece of information is that it takes 'at least' one hour to draw one frame. This leads to the use of the \(\geq\) symbol because the actual time could be equal to or more than a certain amount.

Understanding 'At Least'

The term 'at least' means the value cannot be less than the specified amount. If we need 'at least' 129600 hours to draw the frames, the inequality used to represent this situation is \(H \geq 129600\)—H cannot be less than 129600 hours.

Units of Time Conversion

When working with time-related problems in algebra, it is often necessary to convert units of time to make calculations easier or to align with other units provided in the problem's context. The most common conversions are between hours, minutes, and seconds. Knowing that 1 hour equals 60 minutes and 1 minute equals 60 seconds (or directly that 1 hour equals 3600 seconds) is essential.

For our animation-related problem, it is especially important to convert the length of the film from hours to seconds since the frame rate is given in seconds (frames per second). This conversion is a simple multiplication: for a 1.5-hour movie, you multiply 1.5 by 3600 to obtain the total length in seconds (5400 seconds). To foster a deeper grasp of this concept, practice converting between these units in various directions and using them to solve real-life problems, as this understanding is crucial not only in algebra but also in many practical applications such as cooking, sports, and time management.