Question

the yearly profit $p$ for a widget producer is a function of the number $n$ of widgets sold. the formula is given below.
$p=-180 + 100n-4n^{2}$
here $p$ is measured in thousands of dollars, $n$ is measured in thousands of widgets, and the formula is valid up to a level of 20 thousand widgets sold.
(a) make the graph of $p$ versus $n$.
(b) calculate $p(0)$.
$p(0)=
explain in practical terms what your answer means.
the producer will have a loss if no widgets are sold
the producer will still make a profit if no widgets are sold

Explanation:

Step1: Identify the profit - function

The profit function is given by $P=-180 + 100n-4n^{2}$, where $P$ is in thousands of dollars and $n$ is in thousands of widgets.

Step2: Calculate $P(0)$

Substitute $n = 0$ into the profit - function:
$P(0)=-180+100\times0 - 4\times0^{2}=-180$.

Step3: Interpret the result

Since $P(0)=-180$ (in thousands of dollars), it means that when no widgets are sold ($n = 0$), the producer has a loss of $\$180000$.

Answer:

$P(0)=-180$; The producer will have a loss if no widgets are sold.