A \(4\)-foot tree was planted. The tree grows \(\frac{2}{3}\) of a foot every year. How many years later will the tree reach \(10\) feet?

Since the tree grows \(\frac{2}{3}\) of a foot every year, we can use a table to help write a formula modeling the tree's growth:

Years Passed Tree's Height (ft)
\(0\) \(4\)
\(1\) \(4+\frac{2}{3}\)
\(2\) \(4+\frac{2}{3}\cdot2\)
\(\vdots\) \(\vdots\)
\(y\) \(4+\frac{2}{3}y\)

From this, we've determined that \(y\) years since the tree was planted, the tree's height will be \(4+\frac{2}{3}y\) feet.

To find when the tree will be \(10\) feet tall, we write and solve this equation:

\begin{align*} 4+\frac{2}{3}y\amp=10\\ \multiplyleft{3}\left(4+\frac{2}{3}y\right)\amp=\multiplyleft{3}10\\ 3\cdot4+3\cdot\frac{2}{3}y\amp=30\\ 12+2y\amp=30\\ 2y\amp=18\\ y\amp=9 \end{align*}

Now we will check the solution \(9\) in the equation \(4+\frac{2}{3}y=10\text{:}\)

\begin{align*} 4+\frac{2}{3}y\amp=10\\ 4+\frac{2}{3}(\substitute{9})\amp\stackrel{?}{=}10\\ 4+6\amp\stackrel{\checkmark}{=}10\\ 10\amp\stackrel{\checkmark}{=}10 \end{align*}

In summary, the tree will be \(10\) feet tall \(9\) years later.