$$ 1 \leftarrow \quad \begin{array}{l} ext { The average number of gallons } W ext { of bottled water consumed each year by the consumer can be approximated by } \ W=1.9 d+14.66 ext {, where } d ext { is the number of years since } 2000 ext {. }\end{array} $$

Question

$$ 1 \leftarrow \quad \begin{array}{l}	ext { The average number of gallons } W 	ext { of bottled water consumed each year by the consumer can be approximated by } \ W=1.9 d+14.66 	ext {, where } d 	ext { is the number of years since } 2000 	ext {. }\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's break down and analyze the provided equation:

$$
W = 1.9d + 14.66
$$

**Where:**
- $$ W $$ represents the **average number of gallons of bottled water consumed each year** by a consumer.
- $$ d $$ is the **number of years since the year 2000**.

### Understanding the Equation

1. **Linear Relationship:**
   - The equation is linear, indicating a constant rate of change in bottled water consumption over time.

2. **Slope ($$ 1.9 $$):**
   - This coefficient represents the **rate at which bottled water consumption increases each year**.
   - **Interpretation:** For every additional year after 2000, the average consumption increases by **1.9 gallons**.

3. **Y-Intercept ($$ 14.66 $$):**
   - This is the starting value of $$ W $$ when $$ d = 0 $$ (i.e., in the year 2000).
   - **Interpretation:** In the year 2000, the average consumption was **14.66 gallons**.

### Practical Applications

1. **Calculating Consumption for a Specific Year:**

To find the average consumption in a particular year, determine the value of $$ d $$ by subtracting 2000 from the target year.

**Example:** Calculate $$ W $$ for the year **2010**.
   
   - $$ d = 2010 - 2000 = 10 $$
   - $$ W = 1.9(10) + 14.66 = 19 + 14.66 = 33.66 $$ gallons
   
   **Interpretation:** In 2010, the average bottled water consumption was **33.66 gallons**.

2. **Predicting Future Consumption:**

Suppose you want to estimate the consumption in **2025**.
   
   - $$ d = 2025 - 2000 = 25 $$
   - $$ W = 1.9(25) + 14.66 = 47.5 + 14.66 = 62.16 $$ gallons
   
   **Interpretation:** By 2025, the average consumption is projected to be **62.16 gallons**.

3. **Determining When Consumption Reaches a Certain Level:**

If you want to find out when the average consumption will reach **50 gallons**, solve for $$ d $$:
   
   $$
   50 = 1.9d + 14.66
   $$
   
   $$
   50 - 14.66 = 1.9d
   $$
   
   $$
   35.34 = 1.9d
   $$
   
   $$
   d = \frac{35.34}{1.9} \approx 18.58 \text{ years}
   $$
   
   **Interpretation:** Approximately **18.58 years** after 2000, which would be around the year **2018-2019**, the average consumption reached **50 gallons**.

### Graphical Representation

Plotting $$ W $$ against $$ d $$ would yield a straight line starting at $$ (0, 14.66) $$ with a slope of $$ 1.9 $$. This visualizes the steady increase in bottled water consumption over the years.

### Summary

- **Initial Consumption (2000):** 14.66 gallons
- **Annual Increase:** 1.9 gallons per year
- **Linear Growth:** Predictable and steady increase over time

This model provides a straightforward way to understand and predict trends in bottled water consumption based on the number of years elapsed since 2000.
The average number of gallons of bottled water consumed each year increases by 1.9 gallons for every year after 2000, starting from 14.66 gallons in 2000.

\( 1 \leftarrow \quad \begin{array}{l}\text { The average number of gallons } W \text { of bottled water consumed each year by the consumer can be approximated by } \\ W=1.9 d+14.66 \text {, where } d \text { is the number of years since } 2000 \text {. }\end{array} \)

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