Question

Anyone now how to do this problem I'm lost on it? Please

Answer 1

The profit function P(x) is a parabola.

To find the maximum profit, you need the vertex of the parabola.

The vertex can be found using the formula $x = \frac{-b}{2a}$

where a = -.001 , b = 3,  c = -1800

$x = \frac{-3}{2(-.001)} = 1500$

Part a) Profit is maximized when x = 1500.

To find Profit, plug 'x' into profit function.
$P_{max} = -.001(1500)^2 + 3(1500) - 1800 = 450$

Part b)  Daily max Profit is $450

Next to find the break even point, when profit = 0.

Set profit function equal to 0 and solve for 'x'.

This is a quadratic equation, so use the quadratic formula.

$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a} \\ \\ x = \frac{-3 \pm \sqrt{9 - 4(-.001)(-1800)}}{2(-.001)} \\ \\ x = 829.18 , 2170.82$

Finally, the avg rate of change is simply the slope of the line between 2 points on the parabola.  Use slope formula.

$= \frac{P(2100) - P(1200)}{2100 - 1200} = \frac{90 - 360}{2100-1200} = \frac{-270}{900} = -\frac{3}{10}$

Part d)  Avg rate of change = -3/10