Question

suppose the number of cd players a retail chain is willing to sell per week at a price of $p$ dollars is given by the function $s(p)=\frac{80p}{0.5p + 7}$. find the supply and the instantaneous rate of change of the supply with respect to price when the price is 40 dollars. $s(40)=square$ $s(40)=square$

Explanation:

Step1: Differentiate the supply - function

We have $S(p)=\frac{80p}{0.5p + 7}$. Using the quotient - rule, if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 80p$, $u^\prime=80$, $v = 0.5p + 7$, $v^\prime=0.5$. So, $S^\prime(p)=\frac{80(0.5p + 7)-80p\times0.5}{(0.5p + 7)^{2}}=\frac{40p+560 - 40p}{(0.5p + 7)^{2}}=\frac{560}{(0.5p + 7)^{2}}$.

Step2: Find $S(40)$

Substitute $p = 40$ into $S(p)$: $S(40)=\frac{80\times40}{0.5\times40 + 7}=\frac{3200}{20 + 7}=\frac{3200}{27}\approx118.52$.

Step3: Find $S^\prime(40)$

Substitute $p = 40$ into $S^\prime(p)$: $S^\prime(40)=\frac{560}{(0.5\times40 + 7)^{2}}=\frac{560}{(20 + 7)^{2}}=\frac{560}{729}\approx0.77$.

Answer:

$S(40)=\frac{3200}{27}\approx118.52$, $S^\prime(40)=\frac{560}{729}\approx0.77$