Final answer:
The new unit volume required for Kai to maintain a $500 profit and cover the increased fixed and variable costs, after these costs have risen, is at least 669 copies of the magazine.
Step-by-step explanation:
To calculate the new unit volume required to maintain a $500 profit and cover the increased fixed and variable costs, we need to use the concept where total costs are the sum of fixed plus variable costs. Initially, Kai has a fixed cost of $306 per issue which is to increase by $103, and a variable cost of $0.97 per copy, soon to be increased by $0.40 per copy due to the addition of color.
Let's denote the new number of copies that must be sold to break even as 'n'. We calculate the new fixed costs as $306 + $103 = $409. The new variable cost per copy would be $0.97 + $0.40 = $1.37. To sustain $500 in profit, Kai must account for both these costs and the desired profit within the selling price of the magazine. The following equation arises from these conditions: $2.73n = $409 (new fixed costs) + $1.37n (new variable costs) + $500 (desired profit).
We solve for n, the number of copies that must be sold, to achieve this target: $2.73n - $1.37n = $909 (combining the new fixed cost and desired profit), which simplifies to $1.36n = $909. Dividing both sides by $1.36 gives us n = $909 / $1.36, which computes to approximately 668.4. Since Kai cannot sell a fraction of a magazine, he would need to round up and sell at least 669 copies to cover the increased costs and maintain a $500 profit.