Question

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 y-intercept of 8, 400 and a vertex of (500, 15, 900).
x = number of helmets
y = amount in dollars. the system of equations that represents this situation is y = 26x and
y = -0.030(x - 500)² + 15, 900.
how many helmets must the company make and sell to break even?
635 × helmets
how much will it cost the company to make 600 helmets?
15600 ✔

Explanation:

Step1: Set equations equal (break-even)

$26x = -0.030(x - 500)^2 + 15900$

Step2: Expand the quadratic term

$26x = -0.030(x^2 - 1000x + 250000) + 15900$

Step3: Distribute the coefficient

$26x = -0.030x^2 + 30x - 7500 + 15900$

Step4: Simplify right-hand side

$26x = -0.030x^2 + 30x + 8400$

Step5: Rearrange to standard quadratic form

$0.030x^2 - 4x - 8400 = 0$

Step6: Multiply by 1000 to eliminate decimals

$30x^2 - 4000x - 8400000 = 0$

Step7: Simplify the equation (divide by 10)

$3x^2 - 400x - 840000 = 0$

Step8: Apply quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=3$, $b=-400$, $c=-840000$

$$x=\frac{400\pm\sqrt{(-400)^2-4(3)(-840000)}}{2(3)}$$

Step9: Calculate discriminant

$\sqrt{160000 + 10080000} = \sqrt{10240000} = 3200$

Step10: Solve for positive x

$x=\frac{400 + 3200}{6} = \frac{3600}{6} = 600$
$x=\frac{400 - 3200}{6} = \frac{-2800}{6}$ (discard negative value)

Answer:

600 helmets