total revenue equation TR = 5000Q – 5Q2 and total cost equation C = 500 + 3Q2 Calculate this firm’s profit maximizing price, quantity, total revenue, total cost, and total

Profit maximizing can be achieved using this formula MC=MR.

Differentiate TR to get MR= 5000-10Q
Differentiate TC to get MC=6Q

5000-10Q=6Q >> Q=312.50

Use the Q to get TR and TC.

To find total profit= TR-TC.

To find the profit-maximizing price and quantity for this firm, we need to determine the level of output (Q) that maximizes the profit function. The profit function is calculated as follows:

Profit = Total Revenue – Total Cost

Given the total revenue equation TR = 5000Q – 5Q^2 and the total cost equation C = 500 + 3Q^2, we can calculate the profit function as:

Profit = TR – C => (5000Q – 5Q^2) – (500 + 3Q^2)

Simplifying the equation, we have:

Profit = 5000Q – 5Q^2 – 500 – 3Q^2
= -5Q^2 + 5000Q – 500

To find the profit-maximizing quantity, we take the derivative of the profit function with respect to Q and set it equal to zero.

dProfit/dQ = -10Q + 5000 = 0

Solving for Q, we get:

-10Q + 5000 = 0
-10Q = -5000
Q = -5000 / -10
Q = 500

Now that we know the quantity, we can substitute it back into the total revenue equation to find the total revenue at the profit-maximizing level of output.

TR = 5000Q – 5Q^2
TR = 5000(500) – 5(500^2)
TR = 2,500,000 – 5(250,000)
TR = 2,500,000 – 1,250,000
TR = 1,250,000

To find the total cost, we substitute the profit-maximizing quantity into the total cost equation.

C = 500 + 3Q^2
C = 500 + 3(500^2)
C = 500 + 3(250,000)
C = 500 + 750,000
C = 750,500

Now, let's calculate the profit by subtracting the total cost from the total revenue.

Profit = TR - C
Profit = 1,250,000 - 750,500
Profit = 499,500

Therefore, the profit-maximizing price is not provided in the given equations. The profit-maximizing quantity is 500, the total revenue is 1,250,000, the total cost is 750,500, and the total profit is 499,500.