#### Answer

$y=\frac{3}{2}x+15$

#### Work Step by Step

Every time the $x$-value increases by $1$, the corresponding $y$-value increases by exactly 1.5 feet, every single time. If the general formula for an identity function is $y=mx+b$, then the $m$ value here must be $1.5$, or $\frac{3}{2}$. As for the value of $b$, we can take any $x$-and-$y$-value pair, substitute those $x$-and-$y$-values into the general formula, substitute $m$ out for $1.5$. If we use the first pair (where $x=0$ and $y=15$), then:
$15=1.5\times0+b=0+b=b$
Therefore, $b=15$.
If we remove the pair of $x$-and-$y$-values from the equation, and substitute $15$ as the value of $b$, then
$y=mx+b$
We can confirm that this equation works, by testing if it applies to all the situations in the table.
We already checked that the equation works on the first pair, so we can skip to the second pair, and move forward from there, like this:
$16.5=1.5\times1+15=1.5+15=16.5$
The result is that $16.5=16.5$, which is true, so this pair works. That was just an example, and it goes the same way with the other $x$-and-$y$-pairs, so I won't write all of them out.
As for the final part of the question ("Is it reasonable to assume that the bamboo shoot will continue to grow this way?"), No. I assume that the question intends to ask about the problem's application to the real world, which would mean that the answer is no, because it's unrealistic to assume that a bamboo shoot would continue to grow forever. Eventually, the shoot would have to fall over, or stop growing, or die, and it wouldn't be able to keep up the growth spurt forever.