Maximizing Total Revenue: Finding the Optimal Quantity

What is the value of quantity (Q) that maximizes total revenue?

How can we calculate the optimal quantity (Q) to achieve the highest total revenue?

The value of Q that maximizes total revenue is approximately 38.9.

To find the value of Q that maximizes total revenue, we need to understand that total revenue is equal to the price (P) multiplied by the quantity (Q). In this case, the price is given by the demand function P = (194 - 5Q)^0.5.

To maximize total revenue, we need to find the quantity (Q) that will give us the highest possible value for P * Q.

Step 1: Rewrite the demand function for total revenue

Total revenue (TR) = Price (P) * Quantity (Q)

TR = (194 - 5Q)^0.5 * Q

Step 2: Determine the derivative of the total revenue function

To find the value of Q that maximizes total revenue, we need to take the derivative of the total revenue function with respect to Q.

d(TR)/dQ = (0.5)*(194 - 5Q)^(-0.5) * (-5) + (194 - 5Q)^0.5 * 1

Step 3: Set the derivative equal to zero and solve for Q

To find the value of Q that maximizes total revenue, we need to find the value of Q where the derivative equals zero.

-2.5(194 - 5Q)^(-0.5) + (194 - 5Q)^0.5 = 0

Step 4: Calculate the values of Q

Using a calculator, we can find the values of Q:

Q = (194 - 2.5)/5 ≈ 38.9

Q = (194 + 2.5)/5 ≈ 39.5

Step 5: Determine the value of Q that maximizes total revenue

Comparing the total revenues, we see that the value of Q that maximizes total revenue is approximately 38.9.