Final answer:
To maximize profit using the functions P(x) = 40x - 0.5x^2 and C(x) = 6x + 10, the number of units produced and sold should be 40, resulting in a maximum profit of $800.
Step-by-step explanation:
To calculate the quantity of output that will yield the highest level of profit, we look at the profit function, P(x), which is given by the total revenue function R(x) minus the total cost function C(x). In this scenario, the profit function is P(x) = 40x - 0.5x^2 and the cost function is C(x) = 6x + 10. To find the maximum profit and the number of units that should be produced and sold, we need to calculate the vertex of the profit parabola, which occurs where the derivative of the profit function equals zero.
By taking the derivative of the profit function with respect to x, P'(x)=40-1x, and setting it to zero, we find that x equals 40 units. This is the output level that maximizes profit. To find the maximum profit, we substitute x=40 into the profit function, giving P(40) = 40(40) - 0.5(40)^2, which results in a maximum profit of $800.
Therefore, to achieve the maximum profit, the firm should produce and sell 40 units, resulting in a profit of $800.