A manufacturer can produce digital recorders at a cost of 60 dollars each. It is estimated that if the recorders are sold for P dollars a piece, consumers will buy q= 150-p recorders each month.
What is the average rate of profit obtained as the level of production increases from q=0 to q=15?
I first expressed the manufacturer's profit as a function of q with the formula: P=(150-q)q-60. I thought the next step was to take the derivative and then plug in 0 and 15. That isn't working. Any ideas?
To find the average rate of profit as the level of production increases from q=0 to q=15, we need to calculate the derivative of the profit function with respect to q and then evaluate it at q=0 and q=15.
Let's start by expressing the manufacturer's profit as a function of q: P(q) = (150-q)q - 60.
To find the derivative of P(q), we can use the power rule.
Step 1: Expand the profit function:
P(q) = 150q - q^2 - 60
Step 2: Take the derivative with respect to q:
P'(q) = dP(q)/dq = 150 - 2q
Now we can find the average rate of profit by evaluating P'(q) at q=0 and q=15 and then calculating the difference between the two values divided by the change in q (15-0):
Average rate of profit = (P'(15) - P'(0))/(15-0)
Substituting the values into the equation:
Average rate of profit = (150 - 2(15) - (150 - 2(0))) / 15
Simplifying:
Average rate of profit = (150 - 30 - 150) / 15
Average rate of profit = -30/15
Average rate of profit = -2
Therefore, the average rate of profit obtained as the level of production increases from q=0 to q=15 is -2 dollars per recorder.