Question

A new DVD is available for sale in a store one week after its release. The cumulative revenue, $R, from sales of the DVD in this store in week t after its release is R=f(t)=350 ln tR=f(t)=350lnt with t>1.
Find f(5), f'(5), and the relative rate of change f'/f at t=5. Interpret your answers in terms of revenue.

Answer 1

Solution :

It is given that :

$f'(t) = (350 \ln t)'$

       $=350(\ln t)'$

        $=\frac{350}{t}$

So, $f(5)=350 \ln (5) \approx 563$

     $f'(5) = \frac{350}{5}$

              $=70$

The relative change is then,

$\frac{f'(5)}{f(5)}=\frac{70}{350\ \ln(5)}$

         $=\frac{1}{5\ \ln(5)}$

         $\approx 0.12$

          $=12\%$

This means that after 5 weeks, the revenue from the DVD sales in $563 with a rate of change of $70 per week and the increasing at a continuous rate of 12% per week.