a company plans to sell pens for $2 each. the companys financial planner estimates that the cost, y, of…

Answer:

B. 174

Explanation:

Step1: Write the quadratic - cost function

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex. Given the vertex $(h,k)=(250,370)$, the cost function is $y=a(x - 250)^2+370$. Since the $y$ - intercept is 120 (when $x = 0$), we substitute $x = 0$ and $y = 120$ into the equation: $120=a(0 - 250)^2+370$. Then $120 = 62500a+370$, $62500a=120 - 370=-250$, and $a=-\frac{250}{62500}=-\frac{1}{250}$. So the cost function is $y =-\frac{1}{250}(x - 250)^2+370$.

Step2: Set up the profit equation

The revenue function $R$ from selling $x$ pens at $2$ dollars each is $R = 2x$. The profit function $P$ is $P=R - C$, where $C$ is the cost function. We want to find when $P>0$, so $2x-(-\frac{1}{250}(x - 250)^2+370)>0$. Expand the cost function: $y=-\frac{1}{250}(x^{2}-500x + 62500)+370=-\frac{1}{250}x^{2}+2x - 250 + 370=-\frac{1}{250}x^{2}+2x + 120$. Then the profit inequality becomes $2x-(-\frac{1}{250}x^{2}+2x + 120)>0$, which simplifies to $\frac{1}{250}x^{2}-120>0$. Multiply both sides by 250 to get $x^{2}>30000$.

Step3: Solve for $x$

Take the square - root of both sides of $x^{2}>30000$. Since $x>0$ (number of pens), $x>\sqrt{30000}\approx173.2$. Since the number of pens must be an integer, the minimum number of pens to sell for a profit is $x = 174$.