A manufacturer produces certain devices. The fixed cost of production is $2000. The first unit produced cost $2 and the cost per unit increases by 20 cents or each additional unit produced. What is the total cost C of producing x units?
The premise is faulty. From your data, the first would cost $2 to produce, the second $2.20, the third $2.40, which sounds illogical. Articles can be produced more cheaply on a mass basis.
Assuming you mean that after first one, the remaining units cost $.20 each:
C = $200 + .2(x - 1)
To find the total cost C of producing x units, we need to calculate the sum of the fixed cost and the cost per unit for each additional unit produced.
First, let's calculate the cost per unit. We know that the first unit costs $2, and the cost per unit increases by 20 cents for each additional unit produced. This means the second unit will cost $2 + $0.20 = $2.20, the third unit will cost $2.20 + $0.20 = $2.40, and so on.
Therefore, the cost per unit for any given unit number n can be calculated using the formula: Cost per unit = $2 + ($0.20 * (n - 1))
Next, let's calculate the total cost C of producing x units. We'll be summing up the cost per unit for each unit number from 1 to x.
Total cost C = Fixed cost + (Cost per unit for unit 1 + Cost per unit for unit 2 + ... + Cost per unit for unit x)
Fixed cost = $2000
Now, we can substitute the formula for the cost per unit into the equation for the total cost:
C = $2000 + [($2 + ($0.20 * (1 - 1))) + ($2 + ($0.20 * (2 - 1))) + ... + ($2 + ($0.20 * (x - 1)))]
Simplifying further, we have:
C = $2000 + [($2 * x) + ($0.20 * (1 + 2 + ... + x-1))]
Note that (1 + 2 + ... + x-1) represents the sum of the numbers from 1 to x-1, which can be calculated using the formula for the sum of an arithmetic series: (n * (n-1)) / 2.
So, substituting this formula into the equation for the total cost, we get:
C = $2000 + [($2 * x) + ($0.20 * ((x-1) * x) / 2)]
Simplifying further:
C = $2000 + ($2 * x) + ($0.20 * (x^2 - x) / 2)
Finally, simplifying the equation:
C = $2000 + ($2 * x) + ($0.10 * (x^2 - x))
Therefore, the total cost C of producing x units is given by the equation C = $2000 + ($2 * x) + ($0.10 * (x^2 - x)).