Final answer:
To yield a maximum profit, the manufacturer should produce and sell 80 products. The maximum profit that can be achieved is $40,000.
Step-by-step explanation:
To determine the number of products produced and sold to yield a maximum profit, we need to find the maximum point of the profit function. In this case, the profit function is given as P(x) = -5x^2 + 800x - 15000. The maximum point of the profit function occurs at the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula -b/(2a), where a and b are the coefficients of the quadratic term and the linear term respectively. In this case, a = -5 and b = 800. So, the x-coordinate of the vertex is -800/(2*(-5)) = 80. Therefore, the number of products produced and sold to yield a maximum profit is 80.
To determine the maximum profit, we substitute the x-coordinate of the vertex into the profit function. P(80) = -5(80)^2 + 800(80) - 15000 = 40000. Therefore, the maximum profit is $40,000.