Question

The price-demand and cost functions for the production of microwaves are given as p = 105 - q/90andC(q) = 22000 + 90q,where q is the number of microwaves that can be sold at a price of p dollars per unit and C(q) is the total cost (in dollars) of producing q units(A) Find the profit function in terms of q.P(q) =(B) Evaluate the marginal profit function at q - 1000.P'(1000) =

Answer 1

Part A

$R=(105-\frac{q}{90})^q$$R=105q-\frac{q^2}{90}$$\begin{gathered} Profit=R-C=105q-\frac{q^2}{90}-(22000+90q) \\ \\ p(q)=-\frac{q^2}{90}+15q-22000 \end{gathered}$

Part B

$\begin{gathered} P^{^{\prime}}=-\frac{q}{45}+15 \\ \\ \\ p^{^{\prime}}(1000)=-\frac{1000}{45}+15^2 \\ P^{^{\prime}}(1000)=-22.22222+225 \\ P^{^{\prime}}(1000)=202.777777778 \end{gathered}$

The final answer

Part A

$P(q)=-\frac{q^2}{90}+15q-22000$

Part B

$P^{^{\prime}}(1000)=202.777778$