Solving Tree Height Variation Problem

The problem states that the height of a tree varies as the square root of its age. This can be represented mathematically.

Mathematical Relationship

Let $H$ be the height of the tree and $A$ be its age. The relationship is given by:

$H = k \sqrt{A}$

where $k$ is the constant of proportionality.

Step 1: Calculate the Constant of Proportionality ($k$)

We are given that when the age ($A$) is 324 years, the height ($H$) is 19 feet.

Substitute these values into the equation:

$19 = k \sqrt{324}$

Calculate the square root of 324:

$\sqrt{324} = 18$

Now, solve for $k$:

$19 = k \times 18$

$k = \frac{19}{18}$

Step 2: Calculate the Height at the New Age

We need to find the height ($H$) when the age ($A$) is 81 years, using the calculated constant $k = \frac{19}{18}$.

Use the same formula:

$H = k \sqrt{A}$

Substitute the values $k = \frac{19}{18}$ and $A = 81$:

$H = \frac{19}{18} \sqrt{81}$

Calculate the square root of 81:

$\sqrt{81} = 9$

Now, calculate the height:

$H = \frac{19}{18} \times 9$

Simplify the expression:

$H = \frac{19 \times 9}{18} = \frac{19}{2}$

Convert the fraction to a decimal:

$H = 9.5$

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