Question

the green hat taxi company estimated, on the basis of collected data, that the number of taxi miles driven each day can be modeled by the function ( q = 338l^{0.2} ) when they employ l drivers per day
(a) graph this function for ( 0leq lleq30 )
(b) how many taxi miles are driven each day if there are 9 drivers employed?
(c) does this model indicate that the number of taxi miles increases or decreases as the number of drivers increases?
(d) is the graph of this function concave up or concave down? what does this mean?
(a) choose the correct graph below
(b) if 9 drivers are employed, 525 miles are driven per day
(round to the nearest integer as needed.)
(c) the number of miles driven per day ( square ) as the number of drivers employed increases.

Explanation:

Step1: Analyze the function \(Q = 338L^{0.2}\)

For \(L>0\), the derivative of \(Q\) with respect to \(L\) is \(Q^\prime=\frac{338\times0.2}{L^{0.8}}=\frac{67.6}{L^{0.8}}\). Since \(L>0\), \(Q^\prime>0\).

Step2: Determine the concavity

The second - derivative of \(Q\) with respect to \(L\) is \(Q^{\prime\prime}=\frac{-67.6\times0.8}{L^{1.8}}=\frac{-54.08}{L^{1.8}}\). Since \(L > 0\), \(Q^{\prime\prime}<0\)

Answer:

(c) The number of miles driven per day increases as the number of drivers employed increases.
(d) The graph of the function \(Q = 338L^{0.2}\) is concave down because \(Q^{\prime\prime}<0\). This means that the rate of increase of the number of taxi - miles driven per day is decreasing as the number of drivers increases. That is, adding more drivers still increases the total miles driven, but each additional driver contributes less to the increase in miles than the previous one.