a stereo manufacturer determines that in order to sell x units of a new stereo, the price per unit (in…

a stereo manufacturer determines that in order to sell x units of a new stereo, the price per unit (in dollars) p(x)=1800 - x. the manufacturer also determines that the cost of producing x stereos is given by c(x)=4000 + 20x. what price per unit must be charged to maximize profit? p=$

a stereo manufacturer determines that in order to sell x units of a new stereo, the price per unit (in dollars) p(x)=1800 - x. the manufacturer also determines that the cost of producing x stereos is given by c(x)=4000 + 20x. what price per unit must be charged to maximize profit? p=$

Answer

Explanation:

Step1: Define the revenue function

Revenue $R(x)$ is price per - unit times the number of units sold. Given $p(x)=1800 - x$, then $R(x)=x\cdot p(x)=x(1800 - x)=1800x - x^{2}$.

Step2: Define the profit function

Profit $P(x)$ is revenue minus cost. Given $C(x)=4000 + 20x$, then $P(x)=R(x)-C(x)=(1800x - x^{2})-(4000 + 20x)=-x^{2}+1780x - 4000$.

Step3: Find the derivative of the profit function

The derivative of $P(x)=-x^{2}+1780x - 4000$ using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$ is $P^\prime(x)=-2x + 1780$.

Step4: Set the derivative equal to zero and solve for $x$

Set $P^\prime(x)=0$, so $-2x + 1780 = 0$. Add $2x$ to both sides: $1780 = 2x$. Then $x = 890$.

Step5: Find the price per unit

Substitute $x = 890$ into the price - function $p(x)=1800 - x$. So $p(890)=1800-890 = 910$.

Answer:

$910$