The company's start-up cost is equal to $1/2530.
The company's minimum cost is equal to $15/37,949.
The number of gadgets that this company must produce and sell in order to incur the least cost is equal to 1/30 gadgets.
From the information provided, the cost C (in dollars) for a company to produce and sell x thousand gadgets is modeled by the following function:
Function, C = 1/(30x² − 2x + 2530)
Mathematically, the start-up cost for this company is equal to the total cost when no sales of its gadget has been made i.e the level of activity is at zero (x = 0). Therefore, we would substitute the value of x in the function with zero (0):
Start-up cost = 1/(30(0)² − 2(0) + 2530)
Start-up cost = $1/2530.
First of all, we would determine the number of gadgets that this company must produce and sell in order to incur the least cost by differentiating the function for the total cost using the quotient rule:
dC/dx = [(0(30x² − 2x + 2530) - 1(60x - 2))1/(30x² − 2x + 2530)²]
dC/dx = [(0 - 60x + 2)1/(30x² − 2x + 2530)²]
Equating the marginal cost function to zero, we have:
-60x + 2/(30x² − 2x + 2530)² = 0
-60x + 2 = 0
60x = 2
x = 2/60
x = 1/30 gadgets.
Now, we can determine the minimum cost:
Function, C = 1/(30x² − 2x + 2530)
Minimum cost = 1/(30(1/30)² − 2(1/30) + 2530)
Minimum cost = 1/(1/30 - 1/15 + 2530)
Minimum cost = 1/(-1/15 + 2530)
Minimum cost = 1/(37,949/15)
Minimum cost = $15/37,949
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