One method of calculating the approximate age, in years, of a tree of a particular species is to multiply the diameter of the tree, in inches, by a constant called the growth factor for the species.
The table below gives the growth factors for three species of trees.
$$ \begin{array} {|l|c|} \hline
\hspace{0em}\text{ Species of Tree} & \text{Growth Factor} \\ \hline
\text{American elm} & 4.0 \\ \hline
\text{Black walnut} & 4.5 \\ \hline
\text{Cottonwood} & 2.0 \\ \hline
\end{array} $$
If an american elm and a cottonwood each now have a diameter of 3 feet, which of the following will be closest to the difference, in inches, of their diameters 20 years from now?
(1 foot = 12 inches)
We should first construct a model given the description. Below is the simplist interpretation of the description:
$$ A = dk $$
$$
A = \text{ Age (years)} $$
$$ d = \text{ diameter (inches)} $$
$$k = \text{ growth factor} $$
The question asks us to find the diameters of the trees after 20 years. We can rearrange the equation to find \(d\).
$$ d=\frac{A}{k} $$
American elm
$$ A=20, k=4.0 $$
$$ d=\frac{20}{4} $$
$$ d=5 $$
Cottonwood
$$ A=20, k=2.0 $$
$$ d=\frac{20}{2} $$
$$ d=10 $$
The difference in the diameters is \(10-5=\boxed{5}\). Note that we excluded the current diameter of the trees in our calculation. Because they are equivalent, it does not affect our final calculation.