Final answer:
To find the production level that yields the maximum profit, we need to calculate the difference between the total revenue and total cost at different production levels and find the value of x where the difference is largest. The production level for the maximum profit is 105 units and the maximum profit is $110.25.
Step-by-step explanation:
In order to find the maximum profit, we need to calculate the difference between the total revenue and total cost at different production levels. The total revenue function is given by R(x) = 3x and the total cost function is given by C(x) = 0.01x^2 + 0.9x + 3. To find the production level that yields the maximum profit, we need to find the value of x where the difference between total revenue and total cost is largest.
We can calculate the difference between total revenue and total cost by subtracting the total cost function from the total revenue function: P(x) = R(x) - C(x). Substituting the given functions into this equation, we get P(x) = 3x - (0.01x^2 + 0.9x + 3). Simplifying this equation gives us P(x) = -0.01x^2 + 2.1x.
To find the maximum profit, we need to find the vertex of the parabola represented by the profit function. To do this, we can use the formula x = -b/2a, where a = -0.01 and b = 2.1. Substituting these values into the formula gives us x = -2.1/(2 * (-0.01)) = 105.
Therefore, the production level for the maximum profit is 105 units. To find the maximum profit, we can substitute this value of x into the profit function: P(105) = -0.01(105)^2 + 2.1(105) = $110.25.