When a certain commodity is sold for p dollars per unit, consumers will buy $D ( p ) = \frac { 30,000 } { p }$ units per month. It is estimated that t months from now, the price of the commodity will be $p ( t ) = 0.3 t ^ { 5 / 2 } + 5.4$ dollars per unit. The monthly demand will be decreasing 40 months from now.

Question

Quizplus · Accepted Answer

To determine if the monthly demand will be decreasing 40 months from now, we need to analyze the derivative of $D(p(t))$ with respect to time $t$. First, find $p(t) = 0.3t^{5/2} + 5.4$, then substitute into $D(p) = \frac{30,000}{p}$ to get $D(p(t))$. The derivative of $D(p(t))$ with respect to $t$ will indicate how demand changes over time. If the derivative is negative at $t = 40$, it means demand is decreasing. Given $p(t) = 0.3t^{5/2} + 5.4$, its derivative with respect to $t$ will be positive because the power of $t$ is positive and the coefficient is positive, indicating $p(t)$ increases with $t$. Since $D(p) = \frac{30,000}{p}$, as $p(t)$ increases, $D(p(t))$ decreases because they are inversely related. Therefore, as the price increases over time, the demand decreases, confirming that the monthly demand will be decreasing 40 months from now.

When a Certain Commodity Is Sold for P Dollars Per D(p)=30,000pD ( p ) = \frac { 30,000 } { p }D(p)=p30,000​

When a Certain Commodity Is Sold for P Dollars Per $D ( p ) = \frac { 30,000 } { p }$