Problem 87 - A Graphical Approach to College Algebra

Question

Rainfall By noon, 3 inches of rain had fallen during a storm. Rain continued to fall at a rate of \(\frac{1}{4}\) inch per hour. (a) Find a formula for a linear function \(f\) that models the amount of rainfall \(x\) hours past noon. (b) Find the total amount of rainfall by 2: 30 P.M.

Step-by-Step Solution

Answer

The total rainfall by 2:30 P.M. is 3.625 inches.

Step 1: Understanding the Problem

We are given the amount of rainfall by noon and the rate at which it continues to fall. We need to establish a linear function for the amount of rain as a function of time past noon.

Step 2: Define the Linear Function

Let the total rainfall at time \( x \) hours past noon be \( f(x) \). We know that by noon (i.e., \( x = 0 \)), there had been 3 inches of rain. After that, the rate of rainfall is \( \frac{1}{4} \) inch per hour. The linear function is:\[ f(x) = 3 + \frac{1}{4}x \]

Step 3: Calculate Amount of Rainfall by 2:30 P.M.

2:30 PM is 2.5 hours past noon. Substitute \( x = 2.5 \) into the function:\[ f(2.5) = 3 + \frac{1}{4} \times 2.5 \]

Step 4: Solve for 2:30 P.M. Rainfall

Calculate the value of \( f(2.5) \):\[ f(2.5) = 3 + \frac{1}{4} \times 2.5 = 3 + 0.625 = 3.625 \]

Step 5: Conclusion: Total Rainfall by 2:30 PM

By 2:30 PM, the total amount of rainfall is 3.625 inches.

Key Concepts

Rainfall Modeling

Rainfall modeling with a linear function helps us understand how rainfall accumulates over time. This is especially useful in predicting future rain levels based on a set rate. In our context, we use a linear function to represent the total rainfall in inches. The equation is structured as follows: \( f(x) = 3 + \frac{1}{4}x \).

- **Initial Value:** The constant term "3" is the starting point, representing the rainfall amount by noon.
- **Rate of Change:** The fraction \( \frac{1}{4} \) symbolizes the rate at which rain continues to fall.
- **Variable "x":** It stands for the time in hours past noon.

This simple model allows us to input any time "x" to find out the rain accumulation at that point. Predicting weather-related quantities can thus be easier with these types of straightforward models.

Time Past Noon

Understanding the concept of time past noon is crucial in this rainfall modeling problem. It forms the "independent variable" in our function, represented as "x." Let's break this down further:

- **Starting Point:** Noon is considered \( x = 0 \). All calculations extend from this fixed point.
- **Elapsed Hours:** As time goes on, additional hours past noon are added to "x," which affects the total rainfall number when substituted into our function.

For example, 2:30 P.M. is 2.5 hours past noon. This means you simply count how many hours and fraction of an hour have gone by since exactly 12:00 P.M. Then, include this value in the linear equation. The computation becomes straightforward with this clear frame of reference.

Rate of Change

The rate of change is a fundamental component in linear functions and specifically, it is the "slope" in our rainfall model equation. It tells us how much and how quickly something, like rain, accumulates. Here’s how it works within our function:

- **Amount per Time Unit:** The rate is given by \( \frac{1}{4} \), meaning it rains \( \frac{1}{4} \) of an inch each hour.
- **Uniformity:** The constant rate implies that rain increases steadily, which is characteristic of linear functions.

Using this rate, one can easily calculate how the total rainfall changes with every passing hour. Whenever you have a situation where quantities increase or decrease consistently over time, identifying and utilizing the rate of change is vital for making predictions and drawing conclusions. This makes it an indispensable tool in both theoretical and practical applications.