the table below shows the ages of some trees and their corresponding heights. find an exponential model for…

Explanation:

Step1: Assume exponential model

The general form of an exponential model is $y = ab^x$, where $y$ is the height, $x$ is the age, $a$ and $b$ are constants. We have the following system of equations from the table: When $x = 8$, $y=2.2$, so $2.2 = ab^8$; when $x = 13$, $y = 12.8$, so $12.8=ab^{13}$; when $x = 20$, $y = 29.6$, so $29.6=ab^{20}$. Divide the second - equation by the first - equation: $\frac{ab^{13}}{ab^{8}}=\frac{12.8}{2.2}$, which simplifies to $b^{5}=\frac{12.8}{2.2}\approx5.818$. Then $b=\sqrt[5]{\frac{12.8}{2.2}}\approx1.429$.

Step2: Find the value of $a$

Substitute $b\approx1.429$ into the equation $2.2 = ab^8$. We get $a=\frac{2.2}{b^8}=\frac{2.2}{(1.429)^8}$. $(1.429)^8\approx26.977$, so $a=\frac{2.2}{26.977}\approx0.081$. The exponential model is $y = 0.081\times(1.429)^x$.

Step3: Predict the height of a 30 - year - old tree

Substitute $x = 30$ into the model $y = 0.081\times(1.429)^{30}$. $(1.429)^{30}\approx1379.377$, then $y=0.081\times1379.377\approx111.73$.

Answer:

$111.73$