Question

The number of tree species \(S\) in a given area \(A\) in a forest reserve has been modeled by the power function

\(S\left( A \right) = 0.882{A^{0.842}}\)

where \(A\) is measured in square meters. Find \(S'\left( {100} \right)\)and interpret your answer.

Short answer

The value is \(S'\left( {100} \right) \approx 0.36\).\(S'\left( {100} \right)\)is the rate of change of the number of tree species within every \(100\) square meters of forest area.

Step-by-step solution

Find \(S'\left( A \right)\)

It is given that the number of tree species is given by the function\(S\left( A \right) = 0.882{A^{0.842}}\).Differentiate this function with respect to area\(A\), as shown below:

\(\begin{aligned}S\left( A \right) &= 0.882{A^{0.842}}\\S'\left( A \right) &= \frac{d}{{dA}}\left( {0.882{A^{0.842}}} \right)\\ &= 0.882\left( {0.842{A^{ - 0.158}}} \right)\\ &= 0.742644{A^{ - 0.158}}\end{aligned}\)

Find \(S'\left( {100} \right)\)

Plug\(A = 100\)into\(S'\left( A \right) = 0.742644{A^{ - 0.158}}\)to obtain\(S'\left( {100} \right)\), as shown below:

\(\begin{aligned}S'\left( {100} \right) &= 0.742644{\left( {100} \right)^{ - 0.158}}\\ \approx 0.36\end{aligned}\)

Interpret the result

The derivative \(S'\left( A \right)\) is the rate of change of the number of tree species with respectto change in the forest reserve area. Its units are measured as the number of species per square meter. So, \(S'\left( {100} \right)\)is the rate of change of the number of tree species within every \(100\) square meters of forest area.