The height of a tree varies as the square root of its age. When the age of the tree is 324 years, its height is 19 feet. What will be the height of the tree (in feet) at the age of 81 years?

To find the height of the tree at the age of 81 years, we begin by setting up the mathematical relationship described in the question.

We are given that the height of a tree, which we can represent as h, varies directly as the square root of its age, which we can represent as a. This relationship can be written as:
$h=k\sqrt{a}$
where k is the constant of variation.

First, we use the initial information provided to find the value of k. We know that when the age of the tree is 324 years ($a=324$), the height is 19 feet ($h=19$). Substituting these values into the equation gives:
$19=k\sqrt{324}$

Since the square root of 324 is 18 ($\sqrt{324}=18$), we can simplify the equation:
$19=k·18$

Solving for k, we get:
$k=\frac{19}{18}$

Now, we want to find the height of the tree when its age is 81 years ($a=81$). We substitute the value of k and the new age into our original equation:
$h=\left(\frac{19}{18}\right)\sqrt{81}$

Since the square root of 81 is 9 ($\sqrt{81}=9$), the equation becomes:
$h=\frac{19}{18}·9$

Simplifying the expression:
$h=\frac{19}{2}$
$h=9.5$

Therefore, the height of the tree at the age of 81 years will be 9.5 feet.