Problem 6 - College Algebra with Modeling and Visualization

Question

Time Conversion If \(f(x)\) calculates the number of days in \(x\) hours and \(g(x)\) calculates the number of years in \(x\) days, what does \((g \circ f)\) calculate?

Step-by-Step Solution

Answer

\((g \circ f)(x)\) calculates the number of years in \(x\) hours.

Step 1: Understand the function f(x)

The function \( f(x) \) converts hours into days. Since there are 24 hours in a day, we calculate the number of days as \( f(x) = \frac{x}{24} \).

Step 2: Understand the function g(x)

The function \( g(x) \) converts days into years. Assuming there are 365 days in a year, \( g(x) = \frac{x}{365} \) gives us the number of years in \( x \) days.

Step 3: Determine (g ∘ f)(x)

To find \( (g \circ f)(x) \), we need to apply \( f(x) \) first and then apply \( g(x) \) to the result of \( f(x) \). Specifically, \( (g \circ f)(x) = g(f(x)) = g\left(\frac{x}{24}\right) \).

Step 4: Calculate g(f(x))

Plugging \( f(x) \) into \( g(x) \), we have: \( g\left(\frac{x}{24}\right) = \frac{\frac{x}{24}}{365} = \frac{x}{8760} \). This represents the number of years in \( x \) hours.

Key Concepts

Time Conversion

Time conversion is the process of changing one unit of time into another, such as converting hours into days or days into years. This mathematical problem is very common in everyday scenarios since different activities or events are measured in different time units.
By understanding and applying basic conversion factors, we can easily switch from one unit to another:

• There are 24 hours in a day.
• There are typically 365 days in a year, though leap years have 366 days.

To solve problems involving time conversion, it is crucial to know these fundamental values and how to apply them effectively.

Days to Years

Converting days into years involves determining how many whole years fit into a given number of days. Normally, we assume a year consists of 365 days for simplicity, unless otherwise specified for precision in leap years.
The formula for this conversion is simple; just divide the number of days by 365. That is, if you have a function like \(g(x) = \frac{x}{365}\), it means you are dividing \(x\) days by 365 to get the equivalent number of years.
This approach is typically used in many fields, like calculating the age of somebody in years or determining periods of time for projects or events in years.

Hours to Days

Changing hours into days requires us to know the standard concept that a day is made up of 24 hours. This forms the basis for converting a period of time given in hours into days.
The formula \(f(x) = \frac{x}{24}\) allows us to divide the number of hours by 24 to find the number of days. For example, if you have 48 hours, using this formula yields 48 divided by 24, resulting in 2 days.
So, by applying this straightforward division, anyone can switch between these time units seamlessly, useful for planning and scheduling based on daily cycles.

Mathematical Functions

Mathematical functions are expressions that relate an input to a single output according to some particular rule. In the context of our exercise, when dealing with time conversion between different units, we use functions to represent these transformations.
A composition of functions, like \((g \circ f)\) or \(g(f(x))\), means applying one function (\(f(x)\)) first, and then taking that result as the input for another function (\(g(x)\)). In our example setup, \(f(x)\) converts hours to days, and \(g(f(x))\) converts this result into years.
This layered approach is particularly handy in complex calculations where multiple transformations are involved, allowing us to process input data step-by-step through different stages of conversion.