A pine tree was three 3 meters tall when it was planted. Since planting the tree has grown an equal length each year. 7 years after planting it was 1/9 taller than one year earlier. How tall is the tree 15 years after planting?

Let the height of the tree \( x \) years after planting be denoted by \( h(x) \), and assume that the tree grows at a constant rate of \( r \) meters per year. Given that the tree was 3 meters tall when it was planted, the height function can be written as:

\[
h(x) = 3 + r \cdot x
\]

From the problem, 7 years after planting, the tree was \(\frac{1}{9}\) taller than one year earlier. This gives the relationship:

\[
h(7) = h(6) + \frac{1}{9}h(6)
\]

### Step 1: Write the height at 7 and 6 years after planting.
Using the height function \( h(x) = 3 + r \cdot x \), the heights at 7 and 6 years after planting are:

\[
h(7) = 3 + 7r
\]
\[
h(6) = 3 + 6r
\]

### Step 2: Set up the equation based on the problem condition.
The tree at 7 years is \(\frac{1}{9}\) taller than at 6 years, so:

\[
h(7) = h(6) + \frac{1}{9}h(6)
\]

Substitute the expressions for \( h(7) \) and \( h(6) \):

\[
3 + 7r = (3 + 6r) + \frac{1}{9}(3 + 6r)
\]

### Step 3: Solve for the growth rate \( r \).
First, simplify the right-hand side:

\[
3 + 7r = 3 + 6r + \frac{1}{9}(3 + 6r)
\]

Distribute the \( \frac{1}{9} \):

\[
3 + 7r = 3 + 6r + \frac{3}{9} + \frac{6r}{9}
\]
\[
3 + 7r = 3 + 6r + \frac{1}{3} + \frac{2r}{3}
\]

Now, subtract 3 from both sides:

\[
7r = 6r + \frac{1}{3} + \frac{2r}{3}
\]

To eliminate the fractions, multiply everything by 3:

\[
21r = 18r + 1 + 2r
\]

Combine like terms:

\[
21r = 20r + 1
\]

Subtract \( 20r \) from both sides:

\[
r = 1
\]

### Step 4: Calculate the height of the tree 15 years after planting.
Now that we know the growth rate is \( r = 1 \) meter per year, the height of the tree after 15 years is:

\[
h(15) = 3 + 1 \cdot 15 = 3 + 15 = 18 \text{ meters}
\]

### Final Answer:
The tree will be **18 meters** tall 15 years after planting.

Would you like further explanation on any part? Here are 5 related questions to explore:

1. How would the problem change if the tree had grown by a variable amount each year?
2. What is the significance of the condition that the tree grew \(\frac{1}{9}\) taller in 7 years?
3. How can you generalize the growth of a tree for different growth rates?
4. If the tree stopped growing after a certain number of years, how would you model that scenario?
5. How would the formula change if the tree height followed an exponential growth model?

**Tip:** Always try solving for the variables step-by-step, which helps track down any miscalculations easily.

Learn how to calculate the height of a pine tree 15 years after planting using a linear growth model. This problem involves algebraic equations and proportional relationships, suitable for Grades 8-10. Step-by-step solution provided.

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