ASSUME THAT THE DEMAND FUNCTION IS Q=22-P/5 AND TC=100-10Q+Q^2.

FIND THE OPTIMUM PROFIT
FIND THE MAXIMUM PROFT

To find the optimum profit, we need to find the level of output that maximizes profit. This can be done by calculating the total profit function and finding its maximum value.

Step 1: Calculate the total revenue (TR) function.
The total revenue is calculated by multiplying the quantity sold (Q) by the price (P).
Given that the demand function is Q = 22 - P/5, we can solve for P as follows:
P = 110 - 5Q (by rearranging the demand function)

Now, multiply Q by P to find the total revenue function:
TR = Q * P
TR = Q * (110 - 5Q)

Step 2: Calculate the total cost (TC) function.
The total cost function is given as TC = 100 - 10Q + Q^2.

Step 3: Calculate the profit function.
The profit is calculated as the difference between total revenue and total cost:
Profit (π) = TR - TC
Profit (π) = (Q * (110 - 5Q)) - (100 - 10Q + Q^2)
Simplify and rearrange the equation:
π = 110Q - 5Q^2 - 100 + 10Q - Q^2
π = -6Q^2 + 100Q - 100

Step 4: Find the derivative of the profit function.
To find the maximum profit, we need to find the critical points of the profit function. We can do this by finding the derivative of the profit function and setting it equal to zero.
dπ/dQ = -12Q + 100 = 0

Step 5: Solve for Q to find the level of output that maximizes profit.
-12Q + 100 = 0
-12Q = -100
Q = 8.33 (rounded to 2 decimal places)

Step 6: Substitute the value of Q into the profit function to find the maximum profit.
Profit = -6Q^2 + 100Q - 100
Profit = -6(8.33)^2 + 100(8.33) - 100
Profit = -416.67 + 833.33 - 100
Profit = 316.66 (rounded to 2 decimal places)

Therefore, the optimum profit is approximately 316.66 and the maximum profit occurs when the level of output is approximately 8.33.

To find the optimum profit, we need to find the quantity that maximizes the profit. First, we'll find the revenue function (R) and the cost function (C), then we'll subtract the cost from the revenue to find the profit function (P).

1. Revenue Function (R):
The revenue function is given by the product of the quantity (Q) and the price (P). In this case, the price (P) can be determined by rearranging the demand function. Therefore:
P = 22 - Q/5

Now, we can calculate the revenue function:
R = P * Q
= (22 - Q/5) * Q
= 22Q - Q^2/5

2. Cost Function (C):
The cost function (C) is given as:
C = 100 - 10Q + Q^2

3. Profit Function (P):
The profit function (P) is calculated by subtracting the cost function (C) from the revenue function (R):
P = R - C
= (22Q - Q^2/5) - (100 - 10Q + Q^2)
= 12Q - Q^2/5 - 100

To find the optimum profit, we need to find the quantity (Q) that maximizes the profit function (P). We can find this value by differentiating the profit function with respect to Q and setting it equal to zero.

4. Differentiate the profit function with respect to Q:
dP/dQ = 12 - (2Q)/5

5. Set the derivative equal to zero and solve for Q:
12 - (2Q)/5 = 0
12 = (2Q)/5
60 = 2Q
Q = 30

Now, substitute the value of Q back into the profit function to find the optimum profit:

P = 12Q - Q^2/5 - 100
P = 12 * 30 - (30^2)/5 - 100
P = 360 - 900/5 - 100
P = 360 - 180 - 100
P = 80

Therefore, the optimum profit is 80.