a company plans to sell bicycle helmets for $26 each. the companys business manager estimates that the cost…

a company plans to sell bicycle helmets for $26 each. the companys business manager estimates that the cost, y, of making x helmets is a quadratic function with a y - intercept of 8,400 and a vertex of (500, 15,900).\nx = number of helmets\ny = amount in dollars\nwhich system models this situation?\no y = 26x and y = 8,400(x - 500)^2+15,900\no y = 26x and y = - 0.030(x - 500)^2+15,900\no y = x/26 and y = - 0.030(x - 500)^2+15,900\no y = x/26 and y = 8,400(x - 500)^2+15,900

a company plans to sell bicycle helmets for $26 each. the companys business manager estimates that the cost, y, of making x helmets is a quadratic function with a y - intercept of 8,400 and a vertex of (500, 15,900).\nx = number of helmets\ny = amount in dollars\nwhich system models this situation?\no y = 26x and y = 8,400(x - 500)^2+15,900\no y = 26x and y = - 0.030(x - 500)^2+15,900\no y = x/26 and y = - 0.030(x - 500)^2+15,900\no y = x/26 and y = 8,400(x - 500)^2+15,900

Answer

Explanation:

Step1: Find the revenue function

The company sells each helmet for $26. Revenue $y$ is price per unit times number of units. So the revenue function is $y = 26x$.

Step2: Find the cost - function

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex. Here, the vertex is $(500,15900)$, so $h = 500$ and $k = 15900$, and the function is $y=a(x - 500)^2+15900$. The $y$-intercept is 8400. When $x = 0$, $y=8400$. Substitute $x = 0$ and $y = 8400$ into $y=a(x - 500)^2+15900$: $$8400=a(0 - 500)^2+15900$$ $$8400=a\times250000+15900$$ $$a\times250000=8400 - 15900=- 7500$$ $$a=\frac{-7500}{250000}=-0.030$$ So the cost function is $y=-0.030(x - 500)^2+15900$.

Answer:

$y = 26x$ and $y=-0.030(x - 500)^2+15900$ (the second option)