Answer :
Producing one additional unit at a production level of 10 units would result in a profit loss of $9.
The marginal profit at a given production level is the change in profit that results from producing one additional unit. Mathematically, the marginal profit (P'(x)) can be calculated as follows:
P'(x) = (R(x+1) - R(x)) - (C(x+1) - C(x))
Where R(x) is the revenue at a production level of x units, and C(x) is the cost at a production level of x units.
In this case, the small business has weekly costs of C = 140 + 40x + 8x², where x is the number of units produced each week, and the competitive market price for the business's product is $49 per unit.
The revenue at a given production level can be calculated as the product of the price per unit and the number of units produced:
R(x) = P × x
Where P is the price per unit.
Substituting this expression for R(x) into the equation for P'(x) and simplifying, we get:
P'(x) = (P × (x+1) - P × x) - (C(x+1) - C(x))
= P - (C(x+1) - C(x))
Substituting the expression for C(x) and simplifying, we get:
P'(x) = P - ((140 + 40(x+1) + 8(x+1)²) - (140 + 40x + 8x²))
= P - (40 + 8(2x + 1))
= P - 8x - 48
Therefore, the marginal profit at a given production level x is equal to the price per unit minus 8 times the production level minus $48.
For example, if the production level is x = 10 units, the marginal profit would be:
P'(x) = P - 8x - 48
= $49 - 8(10) - $48
= $-9
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