The company must make and sell either 600 or 466 helmets to break even.
What is cost price?
Cost price is the sum of the costs incurred by a manufacturer to create a specific good or render a specific service.
What is selling price?
The cost a customer pays for a good or service is known as the selling price.
Model for selling price:
y = 26x (given)
Model for cost price:
y = -0.030(x - 500)^2 + 15900 (given)
Equating both the equations:
26x = -0.030(x^2 - 1000x + 250000) + 15900
26x = -0.030x^2 + 30x + 8400
0.030x^2 - 4x - 8400 = 0
x^2 - 133.33x - 280000 = 0
Applying quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
x = (133.33 ± √((133.33)^2 - 4(1)(-280000))) / 2
x = (133.33 ± √1137776.89) / 2
x = (1066 ± 133.33) / 2
x = 600 helmets or x = 466 helmets
Hence, the company should make either 600 or 466 helmets to be neither in profit nor in loss.
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