Each machine at a certain factory can produce 50 units per hour. The setup cost is 45 dollars for each machine and the operating cost is 20 dollars per hour (total, not 20 dollars per machine per hour). You would like to know how many machines should be used to produce 30000 units, with the goal of minimizing production costs.

a) First, find a formula for the total cost in terms of the number of machines, n:

b) The minimum total cost is achieved when using ______ machines for a total cost of _____ dollars.

c = 45n + 20 * 30000/(50n) = 45n + 12000/n

dc/dn = 45 - 12000/n^2
c is minimum when dc/dn = 0

The second and third answers were incorrect. For the second it wants a numerical answer so do I plug in something for n?

a) To find a formula for the total cost in terms of the number of machines, n, we need to consider both the setup cost and the operating cost.

Let's break down the cost into two components:

1. Setup cost: Since the setup cost is 45 dollars per machine and we have n machines, the total setup cost would be 45n dollars.

2. Operating cost: Each machine produces 50 units per hour, so the number of hours required to produce 30000 units would be 30000/50 = 600 hours. The total operating cost would be 20 dollars per hour multiplied by the number of hours, which is 20 * 600 = 12000 dollars.

Total cost = Setup cost + Operating cost = 45n + 12000 dollars.

b) To minimize the total cost, we need to find the value of n that gives the minimum total cost. Since the total cost is a linear function of n, the minimum will occur at the lowest value of n.

To find this value, we can set the derivative of the total cost function equal to zero and solve for n. However, in this case, the total cost function is already in the form of an increasing linear function, so the minimum would occur at the lowest value of n, which is n = 0.

Therefore, the minimum total cost is achieved when using 0 machines, which makes sense since no machines would result in no production cost.

The total cost would be 0 dollars.

a) To find a formula for the total cost in terms of the number of machines, n, we need to consider both the setup cost and the operating cost.

The setup cost is a fixed cost and is multiplied by the number of machines used, n. So, the total setup cost would be 45 dollars times n.

The operating cost is a variable cost that depends on time. Since each machine can produce 50 units per hour and the total number of units needed is 30000, the total time required to produce the units would be 30000 divided by the production rate per hour (which is 50 units per machine). Therefore, the total time required would be 30000 divided by (50n), or 600/n hours.

The operating cost is 20 dollars per hour, so the total operating cost would be 20 dollars per hour times the total time required, which is 20 times (600/n) dollars.

Adding the setup cost and the operating cost gives us the formula for the total cost in terms of the number of machines, n:

Total Cost = Setup Cost + Operating Cost
Total Cost = 45n + (20 * 600/n)
Total Cost = 45n + 12000/n

b) To find the minimum total cost, we need to find the number of machines that minimizes the total cost. We can find this by taking the derivative of the total cost formula with respect to n and setting it equal to zero:

d(Total Cost)/dn = 45 - 12000/n^2

Setting this equal to zero and solving for n:

45 - 12000/n^2 = 0
12000/n^2 = 45
n^2 = 12000/45
n^2 = 266.67
n = sqrt(266.67)
n ≈ 16.33

Since we cannot have fractional machines, we round up to the nearest whole number:

n = ceil(16.33) = 17

So, the minimum total cost is achieved when using 17 machines. To find the actual cost, we substitute this value into the total cost formula:

Total Cost = 45n + 12000/n
Total Cost = 45(17) + 12000/17
Total Cost ≈ 765 + 705.88
Total Cost ≈ 1470.88 dollars

Therefore, the minimum total cost is achieved when using 17 machines for a total cost of approximately 1470.88 dollars.