a manufacturer has daily production costs of C=4000-30x+0.5x where C is the total cost in dollars and x is the number of units produced. (a) How many units should be produced each day to yield a minimum cost? (b) What is the minimum cost?
C has a typo. One of the terms should be x^2, I expect.
If so, then C is just a parabola, and its minimum cost is at the vertex.
To find the number of units that should be produced each day to yield a minimum cost, we need to minimize the cost function (C) with respect to the number of units produced (x).
(a) Finding the number of units:
Step 1: Differentiate the cost function with respect to x.
dC/dx = -30 + 0.5
Step 2: Set the derivative equal to zero and solve for x.
-30 + 0.5x = 0
0.5x = 30
x = 60
Therefore, 60 units should be produced each day to yield a minimum cost.
(b) Finding the minimum cost:
Step 3: Substitute the value of x (60) into the cost function.
C = 4000 - 30(60) + 0.5(60)
C = 4000 - 1800 + 30
C = 2200 + 30
C = 2230
Therefore, the minimum cost is $2230.