A company plans to sell bicycle helmets for \$26 each. The company's business manager estimates that the cost, \( y \), of making \( x \) helmets is a quadratic function with a \( \square y \)-intercept of 8,400 and a vertex of \( (500,15,900) \). \( x= \) number of helmets \( y= \) amount in dollars How many helmets must the company make and sell to break even? \( \square \) \( X \Rightarrow 600 \) helmets COMPLETE How much will it cost the company to make 600 helmets? \$ \( \square \) DONE Which system models this situation? \[ \begin{array}{l} y=26 x \text { and } y=8,400(x-500)^{2}+15,900 \\ y=26 x \text { and } y=-0.030(x-500)^{2}+15,900 \end{array} \]

1. Write the vertex form of the cost function. Since the vertex is \((500,15900)\), the cost function can be written as \[ y=-0.03(x-500)^2+15900 \] where the value of \(a\) is found as follows: - The vertex form is \(\displaystyle y=a(x-500)^2+15900\). - Use the \(y\)-intercept \((0,8400)\): \[ 8400=a(0-500)^2+15900 \quad\Rightarrow\quad 8400=a\cdot250000+15900. \] - Solve for \(a\): \[ a\cdot250000=8400-15900=-7500 \quad\Rightarrow\quad a=-\frac{7500}{250000}=-0.03. \] 2. Determine the break-even point. The company earns revenue by selling helmets at \$26 each. The revenue function is \[ y=26x. \] Break-even occurs when revenue equals cost: \[ 26x=-0.03(x-500)^2+15900. \] It is given that break-even happens at \(x=600\) helmets. We verify: - Revenue at \(x=600\): \[ 26(600)=15600. \] - Cost at \(x=600\): \[ -0.03(600-500)^2+15900=-0.03(100^2)+15900=-0.03(10000)+15900= -300+15900=15600. \] Since both equal \$15,600, break-even occurs when \(600\) helmets are made and sold. 3. Find the cost to make 600 helmets. From the cost function: \[ y=-0.03(600-500)^2+15900=-0.03(100^2)+15900=-300+15900=15600. \] Thus, the cost is \(\$15600\). 4. The system that models this situation is: \[ \begin{cases} y=26x\\[1mm] y=-0.03(x-500)^2+15900 \end{cases} \]