Final answer:
To maximize the profit, we need to find the x-value that maximizes the profit function. By substituting the demand equation into the profit equation and calculating the vertex of the resulting parabola, we find that the daily level of production that will yield a maximum profit for the manufacturer is approximately 6670 rackets/day.
Step-by-step explanation:
To find the daily level of production that will yield a maximum profit for the manufacturer, we need to maximize the profit function using the revenue and cost functions. The revenue function is given by r(x) = px (where p is the price) and the cost function is given by c(x) = 300 + 7x + 0.0003x². The profit function is calculated by subtracting the cost function from the revenue function: p(x) = r(x) - c(x). To find the level of production that maximizes profit, we need to determine the value of x that makes p(x) the highest possible.
Given that the demand equation is p = 11 - 0.0002x, we can substitute this into the profit equation to get p(x) = (11 - 0.0002x)x - (300 + 7x + 0.0003x²). Simplifying, we get p(x) = -0.0003x² - 6.9998x + 11x - 300. Rearranging, we have p(x) = -0.0003x² + 4.0002x - 300.
Now, we need to find the x-value that maximizes the profit function. To do this, calculate the vertex of the parabola represented by the profit function using the formula x = -b/(2a), where a is the coefficient of the x² term and b is the coefficient of the x term. In this case, a = -0.0003 and b = 4.0002. Substituting into the formula, we find that x = -4.0002/(2*-0.0003) = 6670. Therefore, the daily level of production that will yield a maximum profit for the manufacturer is approximately 6670 rackets/day.