Question

small business a small business owner determines that the profit p in dollars from sales of a specific item can be modeled by p = 2x² + 30x, where x is the selling - price of the item in dollars.
a. create a linear - quadratic system to determine the price for which the business will earn $50,000.
b. solve the system in part a to determine the price for which the business will earn $50,000. round to the nearest hundredth, if necessary.
c. what does the solution set of the system mean in the context of the situation?

Explanation:

Step1: Create the quadratic - equation

We know that $P = 2x^{2}+30x$ and $P = 50000$. So the quadratic equation is $2x^{2}+30x=50000$, which can be rewritten as $2x^{2}+30x - 50000=0$. Divide through by 2 to simplify: $x^{2}+15x - 25000=0$.

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For the equation $x^{2}+15x - 25000=0$, we have $a = 1$, $b = 15$, and $c=-25000$. First, calculate the discriminant $\Delta=b^{2}-4ac=(15)^{2}-4\times1\times(-25000)=225 + 100000=100225$. Then $x=\frac{-15\pm\sqrt{100225}}{2}=\frac{-15\pm316.58}{2}$.

Step3: Find the two solutions

$x_1=\frac{-15 + 316.58}{2}=\frac{301.58}{2}=150.79$ and $x_2=\frac{-15 - 316.58}{2}=\frac{-331.58}{2}=-165.79$. Since the selling - price $x$ cannot be negative, we discard $x_2$.

Step4: Interpret the solution set

The solution set $\{150.79\}$ means that when the selling price of the item is approximately $\$150.79$, the business will earn a profit of $\$50000$.

Answer:

a. The quadratic equation is $x^{2}+15x - 25000=0$.
b. The selling price $x\approx150.79$ dollars.
c. The selling price of the item should be approximately $\$150.79$ to earn a profit of $\$50000$.