A small recording studio that produce its own records is able to press 100 records a day for a profit of $3.60 per record. If a more expensive process is used, production can be increased, but the profit per record diminishes by 2 cents for each additional record produced. how many additional records should be produced in order to maximize the total daily profit? What is the total profit?

To determine the optimal number of additional records that should be produced to maximize the total daily profit, we need to analyze the relationship between the profit per record and the number of additional records produced.

Let's denote the number of additional records produced as "x". The profit per record can be expressed as $3.60 - $0.02x, since the profit diminishes by 2 cents for each additional record produced. Therefore, the total daily profit can be calculated as (100 + x) * (3.60 - 0.02x).

To find the value of "x" that maximizes the total daily profit, we can use calculus. We need to find the derivative of the profit function, set it to zero, and solve for "x". The critical points obtained will help us determine the optimal number of additional records.

Let's differentiate the profit function:

Profit = (100 + x) * (3.60 - 0.02x)

Taking the derivative with respect to "x":

d(Profit)/dx = [(100 + x) * (-0.02)] + [(3.60 - 0.02x) * 1]

Simplifying:

d(Profit)/dx = -2 + 100(0.02) + 3.60 - 0.02x

d(Profit)/dx = -2 + 2 + 3.60 - 0.02x

d(Profit)/dx = 3.60 - 0.02x

Setting the derivative to zero:

3.60 - 0.02x = 0

0.02x = 3.60

x = 3.60 / 0.02

x ≈ 180

The critical point is x ≈ 180. However, we need to check whether this value gives us a maximum or minimum. To do that, we can take the second derivative of the profit function.

Taking the second derivative:

d²(Profit)/dx² = -0.02

Since the second derivative is negative (-0.02), this confirms that at x ≈ 180, we have a maximum.

Therefore, the optimal number of additional records to produce to maximize the total daily profit is approximately 180.

To calculate the total daily profit, we can substitute this value of x into the profit function:

Total Profit = (100 + x) * (3.60 - 0.02x)

Total Profit = (100 + 180) * (3.60 - 0.02*180)

Total Profit ≈ 280 * 0.60

Total Profit ≈ $168

Hence, the total daily profit would be approximately $168.