Problem 95 - College Algebra with Modeling and Visualization

Question

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of seconds in \(x\) days

Step-by-Step Solution

Answer

864,000 seconds are in 10 days.

Step 1: Understand the problem

To create a function that calculates the number of seconds in a given number of days, we need to consider how many seconds are in a single day. Since there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, we need to multiply these values together to find the number of seconds in one day.

Step 2: Write the formula for one day

Calculate the number of seconds in one day: \(60 \text{ seconds} \times 60 \text{ minutes} \times 24 \text{ hours} = 86,400 \text{ seconds in one day}\).

Step 3: Define the function g

Now, define the function \(g(x)\) that represents the number of seconds in \(x\) days. Since one day has 86,400 seconds, the function is \(g(x) = 86,400 \times x\).

Step 4: Evaluate g(10)

Substitute \(x = 10\) into the function \(g(x)\): \(g(10) = 86,400 \times 10 = 864,000\).

Step 5: Interpret the result

The evaluation of \(g(10)\) tells us there are 864,000 seconds in 10 days, meaning if you were to count the number of seconds for 10 full days, you would reach 864,000.

Key Concepts

Understanding Functions

A function in mathematics acts like a special rule that gives each input exactly one output. It can be thought of like a machine, where you put in a certain number, and out comes another number as a result of the rule the machine follows. In symbolic representation, functions are often noted by letters such as \(f(x)\) or \(g(x)\), where \(x\) represents the input value.

• The name of the function, like \(g\), identifies the specific rule or formula the function follows.
• The input value, which is \(x\) in \(g(x)\), is a variable that can change.
• The output value depends on the particular rule defined by the function.

In our specific example, the function \(g(x)\) calculates the number of seconds in \(x\) days, turning the input of days into an output of seconds. Every valid input produces one clear output, making functions reliable tools in both mathematics and everyday calculations.

Mathematical Interpretation

When we talk about the mathematical interpretation of functions, we are assigning real-world meanings to the numbers and operations within the formula. Specifically, in this exercise, each part of the function \(g(x) = 86,400 \times x\) illustrates a step in converting days to seconds. Here’s how:

• \(86,400\) represents the total number of seconds in one full day. This is calculated by multiplying the known amounts: 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day.
• The multiplication by \(x\) indicates that we want to extend this conversion rate to however many days we're considering.
• By substituting a particular number for \(x\), like 10, we interpret it as calculating the seconds contained in 10 days.

This reveals that mathematical functions can seamlessly model real-world scenarios, making abstract numbers and symbols meaningful through interpretation. Whether you’re counting seconds in days or any other conversion, understanding these interpretations bridges the gap between numbers and their real applications.

Algebraic Evaluation

Algebraic evaluation involves inputting specific values into a function and solving to find the corresponding output. In our example, the function \(g(x) = 86,400 \times x\) was evaluated using an input value of 10. Here's how the process goes step-by-step:

• Replace the variable \(x\) with 10: \(g(10) = 86,400 \times 10\).
• Perform the multiplication: \(86,400 \times 10 = 864,000\).
• The result, \(864,000\), is the number of seconds in 10 days.

This process shows how algebraic evaluation helps in finding answers for specific cases or quantities, utilizing the formula or model provided by the function. It’s a straightforward method once you understand the basic operations and ensures that mathematical models provide practical, actionable numbers that can be interpreted and used in real-life scenarios.