A popular donut shops uses the following two equations to calculate their revenue and their cost. The cost is calculated using $$ E=0.15 n+6800 $$ where $$ E $$ represents the monthly expenses, and $$ n $$ is the number of donuts that the shop made that month. The revenue is calculated using $$ R=0.85 n $$ where $$ R $$ is the monthly revenue, and $$ n $$ is the number of donuts sold that month. Calculate the monthly break-even point. Round your answer to the next largest donut. $$ n=\square $$ donuts

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The monthly break-even point is 9715 donuts. To find the monthly break-even point, we need to set the revenue equal to the expenses: $$ R = E $$ Substituting the equations for revenue and expenses: $$ 0.85n = 0.15n + 6800 $$ Now, we will solve for $$ n $$: 1. Subtract $$ 0.15n $$ from both sides: $$ 0.85n - 0.15n = 6800 $$ 2. This simplifies to: $$ 0.70n = 6800 $$ 3. Now, divide both sides by $$ 0.70 $$: $$ n = \frac{6800}{0.70} $$ 4. Calculating the right side: $$ n = 9714.2857 $$ Since we need to round to the next largest donut, we round $$ 9714.2857 $$ up to $$ 9715 $$. Thus, the monthly break-even point is: $$ n = 9715 \text{ donuts} $$

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