question 15 (1 point) the demand function for a new cd is p(x) = - 4x + 24 where p(x) represents the selling…

question 15 (1 point) the demand function for a new cd is p(x) = - 4x + 24 where p(x) represents the selling price, in thousands of dollars, and x is the number of cds sold, in thousands. what is the maximum revenue? revenue = p(x)x a) $20 000 b) $3000 c) $24 000 d) $36 000

question 15 (1 point) the demand function for a new cd is p(x) = - 4x + 24 where p(x) represents the selling price, in thousands of dollars, and x is the number of cds sold, in thousands. what is the maximum revenue? revenue = p(x)x a) $20 000 b) $3000 c) $24 000 d) $36 000

Answer

Explanation:

Step1: Recall revenue formula

Revenue $R(x)=x\cdot p(x)$, where $p(x)= - 4x + 24$. So $R(x)=x(-4x + 24)=-4x^{2}+24x$.

Step2: Identify the form of quadratic - function

The function $R(x)=-4x^{2}+24x$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=-4$, $b = 24$, $c = 0$.

Step3: Find the x - value of the vertex

The x - value of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substituting $a=-4$ and $b = 24$ into the formula, we get $x=-\frac{24}{2\times(-4)}=\frac{-24}{-8}=3$.

Step4: Find the maximum revenue

Substitute $x = 3$ into the revenue function $R(x)=-4x^{2}+24x$. Then $R(3)=-4\times3^{2}+24\times3=-4\times9 + 72=-36 + 72 = 36$. Since $x$ is in thousands of CDs and $p(x)$ is in thousands of dollars, the maximum revenue is $$36000$.

Answer:

d) $$36000$