formula 2 points. the table below shows the ages of some trees and their corresponding heights. find an…

Explanation:

Step1: Write the general 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. Using the data points $(x_1,y_1)=(6,2.1)$ and $(x_2,y_2)=(14,5.6)$. Substitute into the equation: For $(6,2.1)$: $2.1 = ab^{6}$ (Equation 1) For $(14,5.6)$: $5.6 = ab^{14}$ (Equation 2)

Step2: Divide Equation 2 by Equation 1

$\frac{ab^{14}}{ab^{6}}=\frac{5.6}{2.1}$. Using the rule of exponents $\frac{b^{m}}{b^{n}}=b^{m - n}$, we get $b^{14 - 6}=b^{8}=\frac{5.6}{2.1}=\frac{8}{3}$. Then $b=\left(\frac{8}{3}\right)^{\frac{1}{8}}\approx1.137$.

Step3: Find the value of $a$

Substitute $b\approx1.137$ into Equation 1: $2.1 = a(1.137)^{6}$. $a=\frac{2.1}{(1.137)^{6}}\approx0.977$. So the exponential model is $y = 0.977\times(1.137)^{x}$.

Step4: Predict the height for a 34 - year - old tree

Substitute $x = 34$ into the model: $y=0.977\times(1.137)^{34}$. $(1.137)^{34}\approx57.977$. $y = 0.977\times57.977\approx56.64$.

Answer:

$56.64$