C(x)=x^2−20x+111 represents the total cost, in dollars, to produce x gallons of paint. How many gallons of paint should be produced to minimize the total cost?

10

To find the number of gallons of paint that should be produced to minimize the total cost, we need to find the minimum point on the cost function.

The cost function is given by:
C(x) = x^2 - 20x + 111

To find the minimum point, we can use calculus. We will take the derivative of the cost function with respect to x and set it equal to zero.

Step 1: Take the derivative of C(x)
C'(x) = 2x - 20

Step 2: Set the derivative equal to zero and solve for x
2x - 20 = 0
2x = 20
x = 20/2
x = 10

The critical point occurs at x = 10. This means that to minimize the total cost, we should produce 10 gallons of paint.

Therefore, to minimize the total cost, 10 gallons of paint should be produced.