Suppose you are the manager of a watch- making firm operating in a competetive marke, your cost of production is given by C=Q-4Q+5Q+100, whe Q is the level of output and C is total cost, the answer thi. If the price of a watch is Br 60, how many watches should you produce to maximize profit?
To maximize profit, we need to determine the level of output that generates the highest profit. In order to do that, we will calculate the profit function and find its maximum value.
1. Begin by calculating the revenue function. Revenue (R) is equal to the product of the price of each watch (P) and the number of watches produced (Q).
R = P * Q
In this case, the price of a watch is given as Br 60. Therefore, the revenue function is:
R = 60Q
2. Next, calculate the cost function. The cost function (C) is given as:
C = Q^2 - 4Q + 5Q + 100
Simplifying the equation:
C = Q^2 + Q + 100
3. To find the profit function (π), subtract the cost function from the revenue function:
π = R - C
π = 60Q - (Q^2 + Q + 100)
π = -Q^2 + 59Q - 100
4. To maximize profit, we need to find the Q value that gives the highest π. The maximum or minimum of a quadratic function occurs at the vertex. The formula for the vertex of a quadratic function of the form ax^2 + bx + c is given by:
Q = -b / 2a
In our case, a = -1, b = 59, and c = -100. Substituting these values, we can find the Q value that maximizes the profit:
Q = -59 / 2(-1)
Q = 29.5
5. Since we cannot produce a fractional quantity of watches, we need the nearest whole number. Therefore, we should produce either 29 watches or 30 watches to maximize profit.
To summarize, if the price of a watch is Br 60 and the cost of production is given by C = Q^2 + Q + 100, we should produce either 29 or 30 watches to maximize our profit.