Question

A textile manufacturer specializing in printed t-shirts has daily production costs given by C = 12,000 – 30x + 0.14x2, where C is the total cost in dollars and x is the number of units produced. How many units should be produced per day to minimize total costs? Round to the nearest whole number. ______ units

Answer 1

To minimize total costs, we need to find the minimum point of the cost function.

By taking the derivative, setting it equal to zero, and solving for x, we can find the number of units that should be produced per day.

Rounding to the nearest whole number, it is approximately 107 units.

Step-by-step explanation:

To find the number of units that should be produced per day to minimize total costs, we need to find the minimum point of the cost function.

The cost function is given by C = 12,000 - 30x + 0.14x^2. We can find the minimum point by taking the derivative of the cost function with respect to x, setting it equal to zero, and solving for x. Let's do that step-by-step:

1. Take the derivative of the cost function: dC/dx = -30 + 0.28x.
2. Set the derivative equal to zero and solve for x: -30 + 0.28x = 0.
3. Add 30 to both sides of the equation: 0.28x = 30.
4. Divide both sides of the equation by 0.28: x = 30/0.28 ≈ 107.14.

Rounding to the nearest whole number, the number of units that should be produced per day to minimize total costs is approximately 107 units.