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Let's analyze the problem and the graph you've provided.

The graph shows the relationship between the **age** of a tree (in years) on the \(x\)-axis and its **height** (in feet) on the \(y\)-axis. It gives one clear point: \((8, 12)\), meaning when the tree is 8 years old, it is 12 feet tall. Additionally, the graph seems to show a straight line through the origin \((0, 0)\), implying a proportional linear relationship.

### 1. Finding the Equation of the Line:

Since it's a straight line, we use the **slope-intercept form** of a linear equation:

\[
y = mx + b
\]

where:
- \(y\) is the height,
- \(x\) is the age,
- \(m\) is the slope,
- \(b\) is the y-intercept (which appears to be 0 from the graph).

#### Step 1: Calculate the Slope \(m\)
The slope \(m\) is calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

From the graph, we can use the points \((0, 0)\) and \((8, 12)\):
- \(x_1 = 0\), \(y_1 = 0\)
- \(x_2 = 8\), \(y_2 = 12\)

\[
m = \frac{12 - 0}{8 - 0} = \frac{12}{8} = 1.5
\]

So the slope \(m = 1.5\).

#### Step 2: Write the Equation
Since the line passes through the origin, the equation simplifies to:

\[
y = 1.5x
\]

### 2. Interpreting the Slope:
The slope of \(1.5\) means that for every year, the tree grows **1.5 feet**. 

### 3. When Will the Tree be 18 Feet Tall?

To find when the tree will be 18 feet tall, set \(y = 18\) in the equation:

\[
18 = 1.5x
\]

Solve for \(x\):

\[
x = \frac{18}{1.5} = 12
\]

So, the tree will be 18 feet tall after **12 years**.

---

Would you like further clarification or details on this? Here are some related questions to deepen understanding:

1. What would happen to the slope if the tree grew at a faster rate?
2. How would the equation change if the line did not pass through the origin?
3. Can you graph another point using the equation \(y = 1.5x\) and verify it?
4. What is the interpretation of the slope when the tree reaches its full height?
5. How would the equation change if the height started from 3 feet at year zero?

**Tip**: Always use clear points from the graph to calculate the slope accurately!

Find the equation and interpret the slope. When will the tree be 18 feet tall?

Learn how to find the equation of a line representing a tree's growth over time and interpret the slope. This math problem uses the slope-intercept form to determine when a tree will be 18 feet tall, with a detailed step-by-step solution. Suitable for Grades 8-10.

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