a firm is planning to manufacture a new product. the sales department estimates the quanity that can be sold depends on the selling price. as the selling price is increased, the quantity that can be sold decreases. they estimate:
P=$35-0.02Q where P=selling price/unit and Q=quantity sold/unit.
on the other hand, management estimates that the average cost of manufacturing and sellling the product will decrease as the quantity sold increases. they estimate:
C=$4Q+$8000.
where C=cost to produce and sell Q/year.
The firm's management wishes to produce and sell the product at the rate that will maximize profit, that is, income minus cost will be a maximum. what quantity should the decision makers plan to produce and sell each year.
I know the answer is 775 units and i have to take derivatives of something because it's asking for a maximum, but how do i put the two equations together.
This is a classic maximize-profits for a monopolist. Always, always, always, maximize where marginal cost (MC) equals marginal revenue (MR).
OK, Total Revenue is P*Q. Using your demand equation TR=35Q-.02Q^2. Marginal revenu is the first derivitive of total revenue, so MR=35-.04Q
Marginal cost is the first derivitive of total cost, so TC = 4.
Maximized profits occur when MC=MR or 4 = 35-.04Q. Solve for Q. (I get 775)
To find the quantity that will maximize profit, we should first express profit as a function of quantity. We can subtract the cost equation from the revenue equation since profit is defined as income minus cost.
Revenue (R) = Selling Price (P) * Quantity (Q)
Cost (C) = $4Q + $8000
Profit (P) = Revenue (R) - Cost (C)
Profit (P) = (35 - 0.02Q)Q - (4Q + 8000)
Simplifying, we have:
P = 35Q - 0.02Q^2 - 4Q - 8000
Now, to find the quantity that maximizes profit, we need to find the derivative of the profit function with respect to quantity (dP/dQ) and set it equal to zero, and solve for Q.
dP/dQ = 35 - 0.04Q - 4 = 0
0.04Q = 35 - 4
0.04Q = 31
Q = 31 / 0.04
Q = 775
Therefore, the quantity that decision-makers should plan to produce and sell each year is 775 units.
To maximize profit, you need to determine the quantity at which the difference between the selling price and the cost is maximized. In this case, the profit function can be expressed as:
Profit = Revenue - Cost
The revenue is equal to the selling price (P) multiplied by the quantity sold (Q), and the cost is given as C. Therefore, the profit function can be written as:
Profit = P * Q - C
Substituting the given equations for P and C:
Profit = (35 - 0.02Q) * Q - (4Q + 8000)
Expanding the equation:
Profit = 35Q - 0.02Q^2 - 4Q - 8000
Simplifying:
Profit = -0.02Q^2 + 31Q - 8000
To maximize profit, you can take the derivative of the profit function with respect to Q and set it equal to zero. Then, solve the resulting equation for Q to find the quantity at which the profit is maximized.
Differentiating the profit function:
dProfit/dQ = -0.04Q + 31
Setting the derivative equal to zero:
-0.04Q + 31 = 0
Solving for Q:
-0.04Q = -31
Q = -31 / -0.04
Q = 775 units
Therefore, to maximize profit, the decision makers should plan to produce and sell 775 units of the product each year.