a company plans to sell embroidered hats for $15 each. the company’s financial planner estimates that the…

Answer:

A. 151

Explanation:

Step1: Write quadratic - function form

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex. Here, $h = 150$ and $k = 9000$, so $y=a(x - 150)^2+9000$. Since the $y$ - intercept is 7920 (when $x = 0$), we substitute $x = 0$ and $y = 7920$ into the equation: $7920=a(0 - 150)^2+9000$.

Step2: Solve for $a$

$7920=a\times22500 + 9000$. $a\times22500=7920 - 9000=-1080$. $a=\frac{-1080}{22500}=-0.048$. So the cost function is $y=-0.048(x - 150)^2+9000$.

Step3: Set up profit equation

The revenue function $R$ is $R = 15x$ (since each hat is sold for $15$). The company makes a profit when $R>y$, so $15x>-0.048(x - 150)^2+9000$. Expand $-0.048(x - 150)^2=-0.048(x^{2}-300x + 22500)=-0.048x^{2}+14.4x - 1080$. The inequality becomes $15x>-0.048x^{2}+14.4x - 1080+9000$. $0.048x^{2}+0.6x - 7920>0$. Divide through by 0.048: $x^{2}+12.5x - 165000>0$.

Step4: Solve quadratic inequality

For the quadratic equation $x^{2}+12.5x - 165000 = 0$, the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b = 12.5$, and $c=-165000$. $x=\frac{-12.5\pm\sqrt{(12.5)^{2}-4\times1\times(-165000)}}{2\times1}=\frac{-12.5\pm\sqrt{156.25 + 660000}}{2}=\frac{-12.5\pm\sqrt{660156.25}}{2}$. $x=\frac{-12.5\pm812.5}{2}$. We get $x_1=\frac{-12.5 + 812.5}{2}=\frac{800}{2}=400$ and $x_2=\frac{-12.5 - 812.5}{2}=\frac{-825}{2}=-412.5$. Since the number of hats $x>0$, the company makes a profit when $x>400$. The minimum number of hats to make a profit is 401.