The marketing department of a company estimates that the demand for a product is given by $p = 180 - 0.0001 x,$ where $p$ is the price per unit and $x$ is the number of units. The cost $C$ of producing $x$ units is given by $C = 450,000 + 50 x,$ and the profit $P$ for producing and selling $x$ units is given by $P = R - C = x p - C.$ Sketch the graph of the profit function and estimate the number of units that would produce a maximum profit.
A)690,000 units
B)650,000 units
C)610,000 units
D)710,000 units
E)580,000 units

Question

Quizplus · Accepted Answer

To find the maximum profit, we first express the profit function $P$ in terms of $x$: $P = x(180 - 0.0001x) - (450,000 + 50x)$. Simplifying, $P = 180x - 0.0001x^2 - 450,000 - 50x$, or $P = -0.0001x^2 + 130x - 450,000$. This is a quadratic function opening downwards, and its maximum occurs at the vertex. The vertex $x$ value is given by $-b/(2a)$, where $a = -0.0001$ and $b = 130$. Calculating gives $x = 650,000$ units.

The Marketing Department of a Company Estimates That the Demand p=180−0.0001x,p = 180 - 0.0001 x,p=180−0.0001x,

The Marketing Department of a Company Estimates That the Demand $p = 180 - 0.0001 x,$