###### 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.

in-context