A firm finds that it can sell all the radios it manufactures st a price of $75 each. If x radios are manufactured each day and C(x)=x^2 +25x +100 is the daily total cost of production, how many radios should be produced each day for the firm to maximize daily profit? What is the maximum daily profit?

To find the number of radios that should be produced each day to maximize daily profit, we need to determine the derivative of the profit function with respect to the number of radios produced and set it equal to zero.

Let's start by determining the profit function. Profit is given by the difference between revenue and cost.

Revenue = Price * Quantity
Revenue = $75 * x

Cost = C(x) = x^2 + 25x + 100

Profit = Revenue - Cost
Profit = (75 * x) - (x^2 + 25x + 100)
Profit = 75x - x^2 - 25x - 100
Profit = -x^2 + 50x - 100

To find the number of radios that should be produced each day for maximum daily profit, we find the derivative of the profit function:

d(Profit)/dx = -2x + 50

Now, set the derivative equal to zero and solve for x:

-2x + 50 = 0
-2x = -50
x = 25

Therefore, to maximize daily profit, the firm should produce 25 radios each day.

To find the maximum daily profit, substitute the value of x into the profit function:

Profit = -x^2 + 50x - 100
Profit = -(25)^2 + 50(25) - 100
Profit = -625 + 1250 - 100
Profit = 525

Therefore, the maximum daily profit is $525.

To recap:
- The firm should produce 25 radios each day to maximize daily profit.
- The maximum daily profit is $525.