Suppose the cost C(q) (in dollars) of producing a quantity q of a product equals
C(q) = 500 + 2q +1/5q2.
The marginal cost MC(q) equals the instantaneous rate of change of the total cost.
Find the marginal cost when a quantity of 10 items are being produced.
assume you mean:
C(q) = 500 + 2q +(1/5)q^2
dC/dq = 2 + (1/5) 2 q
= 2 + (2/5) q
when q = 10 this is
2+4
= 6 = cost of 11th item
To find the marginal cost when a quantity of 10 items are being produced, we need to find the derivative of the cost function C(q) with respect to quantity q.
Given: C(q) = 500 + 2q + (1/5)q^2
To find the derivative, we differentiate each term:
dC/dq = d(500)/dq + d(2q)/dq + d((1/5)q^2)/dq
Since the derivative of a constant is 0, the first term becomes 0:
dC/dq = 0 + 2 + (1/5) * d(q^2)/dq
Using the power rule, the derivative of q^2 is 2q:
dC/dq = 2 + (1/5) * 2q
Simplifying the expression:
dC/dq = 2 + (2/5)q
Now, substitute q = 10 into the equation to find the marginal cost when 10 items are being produced:
MC(10) = 2 + (2/5) * 10
MC(10) = 2 + (2/5) * 10
MC(10) = 2 + 4
MC(10) = 6
Therefore, the marginal cost when a quantity of 10 items are being produced is $6.
To find the marginal cost when a quantity of 10 items is being produced, we need to find the derivative of the cost function with respect to q, and then evaluate it at q = 10.
Let's differentiate the cost function C(q) = 500 + 2q + (1/5)q^2 with respect to q:
dC/dq = d(500 + 2q + (1/5)q^2)/dq
To differentiate this expression, we can apply the power rule for derivatives:
dC/dq = 0 + 2 + (1/5)(2q)
Simplifying, we get:
dC/dq = 2 + (2/5)q
Now, we can evaluate this derivative at q = 10:
MC(10) = 2 + (2/5)(10)
= 2 + (2/5)(10)
= 2 + (2/5)(10)
= 2 + 4
= 6
Therefore, the marginal cost when producing a quantity of 10 items is $6.