###### Example3.2.2

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.