a company plans to sell embroidered hats for $15 each. the companys 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 $y=a(x - 150)^2+9000$.

Step2: Find the value of $a$

Since the $y$ - intercept is 7920 (when $x = 0$), we substitute $x = 0$ and $y = 7920$ into the equation: $7920=a(0 - 150)^2+9000$. Then $7920 = 22500a+9000$, $22500a=7920 - 9000=-1080$, and $a=-\frac{1080}{22500}=-\frac{6}{125}$. So the cost function is $y =-\frac{6}{125}(x - 150)^2+9000$.

Step3: Set up profit equation

The revenue function $R(x)=15x$. The company makes a profit when $R(x)>y$, so $15x>-\frac{6}{125}(x - 150)^2+9000$.

Step4: Expand and simplify

Expand $-\frac{6}{125}(x - 150)^2=-\frac{6}{125}(x^{2}-300x + 22500)=-\frac{6}{125}x^{2}+\frac{1800}{125}x-\frac{135000}{125}$. The inequality becomes $15x>-\frac{6}{125}x^{2}+\frac{1800}{125}x-\frac{135000}{125}+9000$. Multiply through by 125 to clear the fraction: $1875x>-6x^{2}+1800x - 135000 + 1125000$. Rearrange to get $6x^{2}+75x-990000>0$, or $x^{2}+12.5x - 165000>0$.

Step5: Solve the quadratic inequality

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