Question

the profit formula for a company can be represented using the function $p(x) = -30x^2 + 1,700x - 15,000$, where $x$ is the price of the product. use a graphing calculator to determine the price greater than the price at the maximum point of the function, when the profit is $6,830. round to the nearest cent if necessary. enter your answer in the box. $\square$

Explanation:

Step1: Find max price vertex

For quadratic $ax^2+bx+c$, vertex $x=-\frac{b}{2a}$.
$x = -\frac{1700}{2(-30)} = \frac{1700}{60} \approx 28.33$

Step2: Set profit to $6830$

Set $P(x)=6830$, solve $-30x^2+1700x-15000=6830$.
Rearrange: $-30x^2+1700x-21830=0$
Multiply by $-1$: $30x^2-1700x+21830=0$

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, $a=30,b=-1700,c=21830$.
Discriminant: $\Delta=1700^2-4(30)(21830)=2890000-2619600=270400$
$\sqrt{\Delta}=520$
$x=\frac{1700\pm520}{60}$
$x_1=\frac{1700+520}{60}=\frac{2220}{60}=37$, $x_2=\frac{1700-520}{60}=\frac{1180}{60}\approx19.67$

Step4: Select price > max point

Choose $x=37$ (since $37>28.33$)

Answer:

$\$37.00$