I do not have any Idea how to set this up. Please walk me through the steps.
A.) If a customer buys x copies then he or she pays $500 sqrt x. it cost the company $10,000 to develop the program and $2 to manufacture each copy. If a single customer were to buy all the copies of Dogwood software how many copies would the customer have to buy for the software companies average profit per copy to maximized? How are average profit and marginal profit related at this number of copies?
B.) repeat presiding exercise with the charge to the customer $600 sqrt x and the cost to develop the program $9,000.
A) To find the number of copies the customer would have to buy for the software company's average profit per copy to be maximized, we need to calculate the average profit and marginal profit equations.
1. Average Profit (AP) per copy:
The company's average profit per copy is calculated by subtracting the cost per copy (manufacturing cost) from the price per copy (customer charge) and dividing it by the number of copies:
AP(x) = (Customer charge - Manufacturing cost) / x
2. Marginal Profit (MP):
The company's marginal profit is the derivative of the average profit function with respect to the number of copies:
MP(x) = d(AP(x)) / dx
To maximize the average profit, we need to find the number of copies that maximizes the derivative of the average profit function (MP(x) = 0).
Taking the specified values into account:
- The price per copy (customer charge) is $500 sqrt(x) for the first case and $600 sqrt(x) for the second case.
- The cost to develop the program is $10,000 for the first case and $9,000 for the second case.
- The manufacturing cost is $2 per copy in both cases.
Let's calculate the number of copies (x) and the relationship between average profit and marginal profit for each case:
A) Customer charge: $500 sqrt(x)
Cost to develop the program: $10,000
Manufacturing cost: $2
1. Calculate the average profit equation (AP(x)):
AP(x) = (500 sqrt(x) - 2) / x
2. Calculate the marginal profit equation (MP(x)):
MP(x) = d(AP(x)) / dx
3. Find x when MP(x) = 0:
Set MP(x) = 0 and solve for x to find the number of copies that maximize average profit.
B) Customer charge: $600 sqrt(x)
Cost to develop the program: $9,000
Manufacturing cost: $2
1. Calculate the average profit equation (AP(x)):
AP(x) = (600 sqrt(x) - 2) / x
2. Calculate the marginal profit equation (MP(x)):
MP(x) = d(AP(x)) / dx
3. Find x when MP(x) = 0:
Set MP(x) = 0 and solve for x to find the number of copies that maximize average profit.
Please let me know if you would like me to calculate the specific equations or help you find the optimal number of copies for each case.
A.) To determine the number of copies the customer would have to buy for the software company's average profit per copy to be maximized, we need to find the point where the marginal profit equals zero.
1. Start by calculating the total profit function. The profit earned per copy is the difference between the price paid by the customer and the cost to manufacture each copy. Therefore, the profit function is:
Profit(x) = (500√x - 2)x - 10,000
2. Find the marginal profit function by taking the derivative of the profit function with respect to x:
Marginal Profit(x) = d(Profit)/dx = (500√x - 2) + x(250/√x)
3. Set the marginal profit function equal to zero and solve for x:
(500√x - 2) + x(250/√x) = 0
4. Simplify the equation and solve for x. This may involve squaring both sides of the equation to eliminate the square root:
500√x - 2 + 250√x = 0
750√x = 2
√x = 2/750
x = (2/750)^2
5. Calculate the value of x using a calculator:
x ≈ 0.000005333
So, the customer would have to buy approximately 0.000005333 copies of the software for the average profit per copy to be maximized.
Regarding the relationship between average profit and marginal profit at this number of copies:
- At this number of copies, the average profit reaches its maximum value.
- When the marginal profit is zero, it means that the profit is neither increasing nor decreasing. Therefore, at this point, the average profit is maximized.
B.) The steps for solving this part of the exercise would be similar to the ones explained above, with adjustments made for the different values.
1. Calculate the new profit function using the given values:
Profit(x) = (600√x - 2)x - 9,000
2. Find the marginal profit function by taking the derivative of the profit function with respect to x:
Marginal Profit(x) = d(Profit)/dx = (600√x - 2) + x(300/√x)
3. Set the marginal profit function equal to zero and solve for x:
(600√x - 2) + x(300/√x) = 0
4. Simplify the equation and solve for x.
Solve this equation to find the value of x.
5. Finally, calculate the value of x using a calculator.
By following these steps, you can determine the number of copies the customer would have to buy for the software company's average profit per copy to be maximized for both scenarios.