To determine the optimal integer output level, we need to consider the relationship between the revenue and the cost of producing the product.

Step 1: Define the key components

1. Revenue (R):
   Since each unit sells for $43, the revenue from selling $q$ units is: $R(q) = 43q$

2. Marginal Cost (MC):
   The marginal cost for producing the $q$th unit is given by: $MC(q) = 8q$ The total cost (C) of producing $q$ units is the sum of the marginal costs for each unit plus the fixed costs: $C(q) = \text{Fixed Costs} + \sum_{i=1}^{q} 8i$ The fixed cost is given as $350.

3. Total Cost (C): The sum of the marginal costs up to $q$ units is the sum of an arithmetic series: $\sum_{i=1}^{q} 8i = 8 \left( \frac{q(q+1)}{2} \right)$ Therefore, the total cost function becomes: $C(q) = 350 + 4q(q+1)$

Step 2: Define the profit function

The profit function $\Pi(q)$ is the revenue minus the total cost: $\Pi(q) = R(q) - C(q) = 43q - \left(350 + 4q(q+1)\right)$ Simplifying this: $\Pi(q) = 43q - 350 - 4q^2 - 4q$ $\Pi(q) = -4q^2 + 39q - 350$

Step 3: Optimize the profit function

To find the optimal output level, we need to maximize the profit function $\Pi(q)$. Since this is a quadratic function, we can find the maximum by taking the derivative with respect to $q$, setting it equal to zero, and solving for $q$.

Take the derivative of $\Pi(q)$ with respect to $q$: $\frac{d\Pi(q)}{dq} = -8q + 39$ Set this equal to zero to find the critical point: $-8q + 39 = 0$ $8q = 39$ $q = \frac{39}{8} = 4.875$ Since $q$ must be an integer, we check the profit at $q = 4$ and $q = 5$.

Step 4: Evaluate the profit at $q = 4$ and $q = 5$

Calculate the profit at $q = 4$: $\Pi(4) = -4(4)^2 + 39(4) - 350 = -64 + 156 - 350 = 92 - 350 = -258$

Calculate the profit at $q = 5$: $\Pi(5) = -4(5)^2 + 39(5) - 350 = -100 + 195 - 350 = 95 - 350 = -255$

Since the profit is higher at $q = 5$, the optimal integer output level is $q = 5$.

Conclusion

The optimal integer output level is 5 units.

Do you want more details or have any questions?

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1. What is the impact of increasing fixed costs on the optimal output level?
2. How does a change in the unit selling price affect the optimal output?
3. What would happen to the profit if the selling price was lower than the marginal cost of the first unit?
4. How can we calculate the break-even point for this product?
5. How does the quadratic nature of the profit function influence the optimal output level?
6. What is the effect of economies of scale on the marginal cost and optimal output?
7. How would the optimal output level change if the marginal cost function was linear but with a different slope?
8. What are the implications of producing at a non-optimal output level?

Tip: Always verify if the calculated optimal output level yields a higher profit by comparing it with nearby integer values.