Final answer:
The rate of change of profit with respect to the number of toasters produced when x = 100 is $14 per toaster, calculated by taking the derivative of the profit function and substituting x = 100.
Step-by-step explanation:
To calculate the rate of change of profit with respect to the number of toasters produced when x = 100, we need to find the derivative of the profit function P(x) with respect to x and then substitute x = 100 into the derivative. The profit function given is P(x) = -0.05x² + 24x - 240. The first derivative of this function, P'(x), represents the rate of change of profit and is calculated as follows:
P'(x) = d/dx (-0.05x² + 24x - 240)
P'(x) = -0.1x + 24
Now we substitute x = 100 into the derivative to find the rate of change of profit at this production level:
P'(100) = -0.1(100) + 24
P'(100) = -10 + 24
P'(100) = 14
Therefore, the rate of change of profit with respect to the number of toasters produced when x = 100 is $14 per toaster.
Complete Question :
The profit of manufacturing "x" toasters in one day is given by the following: "P(x) = -0.05x² + 24x - 240," where "x" represents the number of toasters produced. Calculate the rate of change of profit with respect to "x" when "x = 100."