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.