Explanation:
R(x) = -x² + 10x + 30000
R(x) is the revenue ("winning", sales price minus costs), x is the internal cost of the batteries.
a) the break even point is, where the revenue is 0 (sales income and cost are equal, so the company is at the brink to get a positive result)
so, we have
0 = -x² + 10x + 30000
or (after multiplying everything by -1)
0 = x² - 10x - 30000
the general solution to such a quadratic equation
0 = ax² + bx + c
is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = -10
c = -30000
x = (10 ± sqrt((-10)² - 4×1×-30000))/(2×1) =
= (10 ± sqrt(100 + 120000))/2 =
= (10 ± sqrt(120100))/2 =
= (10 ± 346.554469...)/2
as negative costs don't make sense in this regard, we are only looking at the plus solution :
x = (10 + 346.554469...)/2 = 356.554469.../2 =
= 178.2772345... ≈ $178.28
the other one would be
x = (10 - 346.554469...)/2 = -336.554469.../2 =
= -168.2772345... ≈ -$168.28
the break even point would be at costs of $178.28.
anything below that threshold would increase the revenue. anything above that threshold would create a negative revenue (loss).
the theoretical other break even point would be at -$168.28. it would mean, whenever they buy or manufacture the batteries, they would get money (instead of having to pay for it). that is not realistic. and funnily, the more above this threshold (until 0), the bigger the revenue.
b)
there is no limit. x can go as high as theoretically possible (mathematically to infinity).
the maximum cost to at least not make a loss is the break even point of a).
c)
the maximum revenue is when there are no costs for the batteries (costs x = 0).
the max. revenue is then
R(0) = -0² + 10×0 + 30000 = $30,000