Question

the number of tree species s in a given area a in a forest reserve has been modeled by the power function s(a)=0.882a^{0.842} where a is measured in square meters. find s(106). (round your answer to five decimal places.) s(106)= species/m^{2} interpret your answer. the rate of change of the number of tree species with respect to area is -0.35545 species/m^{2}. the rate of change of the number of tree species with respect to area is 0.35545 species/m^{2}. the rate of change of the number of tree species with respect to area is -44.74826 species/m^{2}. the rate of change of the number of tree species with respect to area is 44.39281 species/m^{2}. the rate of change of the number of tree species with respect to area is 44.74826 species/m^{2}.

Explanation:

Step1: Differentiate the power - function

Use the power - rule for differentiation $\frac{d}{dA}(x^n)=nx^{n - 1}$. Given $S(A)=0.882A^{0.842}$, then $S'(A)=0.882\times0.842A^{0.842 - 1}=0.742644A^{-0.158}$.

Step2: Evaluate $S'(A)$ at $A = 106$

Substitute $A = 106$ into $S'(A)$. So $S'(106)=0.742644\times(106)^{-0.158}$.
We know that $a^{-n}=\frac{1}{a^{n}}$, so $S'(106)=0.742644\times\frac{1}{106^{0.158}}$.
Calculate $106^{0.158}\approx2.08937$.
Then $S'(106)=\frac{0.742644}{2.08937}\approx0.35545$.

Answer:

$S'(106)=0.35545$ species/m²
The rate of change of the number of tree species with respect to area is 0.35545 species/m².