Final answer:
The break-even points are found by setting the total cost function equal to the total revenue function and solving for x. After simplification, this results in a quadratic equation that we can solve using the quadratic formula or factorization to get the break-even quantities.
Step-by-step explanation:
To find the break-even points where total costs c(x) equal total revenues r(x), we need to set c(x) = r(x) and solve for x. Given the cost function c(x) = 35,000 + 55x + 0.3x² and the revenue function r(x) = 625x - 0.7x², equate them and solve the resulting quadratic equation:
35,000 + 55x + 0.3x² = 625x - 0.7x²
Combine like terms:
0.3x² + 55x + 35,000 = 625x - 0.7x²
Rearrange:
x² + 570x - 35,000 = 0
Now, apply the quadratic formula or factorization to find the values of x that satisfy the equation, which will give you the break-even quantities. Note that for real-life scenarios, we usually consider only the positive root of the equation as it makes sense in the context of production and sales.