The marginal cost function is given by dc/dq=0.08q^2-1.8q+2.5 where c is the total cost in producing q units of product. The fixed cost is 5000. Find the total cost for producing 1000 units
dc/dq=0.08q^2-1.8q+2.5
c = (.08/3) q^3 - .9 q^2 + 2.5q + 5000
plug in 1000 for q
To find the total cost for producing 1000 units, we need to integrate the marginal cost function and add the fixed cost.
The marginal cost function is given as:
dc/dq = 0.08q^2 - 1.8q + 2.5
To integrate, we treat the marginal cost function as a standard polynomial and use the power rule. Integrate each term separately:
∫(0.08q^2 - 1.8q + 2.5) dq
= 0.08 ∫(q^2) dq - 1.8 ∫(q) dq + 2.5 ∫(1) dq
= 0.08 (q^3/3) - 1.8 (q^2/2) + 2.5q + C
where C is the constant of integration.
Since we're interested in the total cost for producing 1000 units, we can substitute q = 1000 into the integrated function. Let's find the value of the constant of integration first.
Given that the fixed cost is $5000, we can substitute q = 0 into the integrated function to find C:
C = 0.08 (0^3/3) - 1.8 (0^2/2) + 2.5(0) + 5000
C = 0 - 0 + 0 + 5000
C = 5000
Now, substitute q = 1000 and C = 5000 into the integrated function:
Total cost = 0.08(1000^3/3) - 1.8(1000^2/2) + 2.5(1000) + 5000
= 0.08(1000^3)/3 - 1.8(1000^2)/2 + 2500 + 5000
= (80,000,000/3) - (9,000,000/2) + 2500 + 5000
= 26,666,666.67 - 4,500,000 + 2500 + 5000
= 22,266,166.67
Therefore, the total cost for producing 1000 units is $22,266,166.67.