When a certain commodity is sold for p dollars per unit,consumers will buy $D ( p ) = \frac { 31,500 } { p }$ units per month.It is estimated that t months from now,the price of the commodity will be $p ( t ) = t ^ { 2 / 3 } + 5.15$ dollars per unit.The approximate rate at which the monthly demand will be changing with respect to time in 27 months is
A)-32 units per month
B)-35 units per month
C)-132 units per month
D)35 units per month

Question

Quizplus · Accepted Answer

To find the rate at which the monthly demand is changing with respect to time, we need to find $\frac{dD}{dt}$. Using the chain rule, $\frac{dD}{dt} = \frac{dD}{dp} \cdot \frac{dp}{dt}$. First, find $\frac{dD}{dp} = -\frac{31,500}{p^2}$. Then, find $\frac{dp}{dt} = \frac{2}{3}t^{-1/3}$. At $t = 27$, $p(27) = 27^{2/3} + 5.15 = 8 + 5.15 = 13.15$. $\frac{dp}{dt} = \frac{2}{3} \cdot 27^{-1/3} = \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9}$. Therefore, $\frac{dD}{dt} = -\frac{31,500}{13.15^2} \cdot \frac{2}{9} \approx -35$ units per month.

When a Certain Commodity Is Sold for P Dollars Per D(p)=31,500pD ( p ) = \frac { 31,500 } { p }D(p)=p31,500​

When a Certain Commodity Is Sold for P Dollars Per $D ( p ) = \frac { 31,500 } { p }$