the marginal cost function for widgets is
dr/dq= 0.001Q^2+0.01Q+10
total fixed xosts equal $500
part 1= convert the marginal cost function into total cost function.
part 2= determine total costs when Q=100.
r = (1/3)(.001)Q^3 + (1/2)(.01)Q^2 + 10 Q + 100
but where did you the "1/2 and 1/3" from?
Oh my
Have you done integration of polynomials?
integral x^n dx = 1/(n+1)x^(n+1)
so
integral x^2 dx = (1/3)x^3
no I am trying to teach myself. however would the answer be 3843.33
if Q = 100 = 10^2
r = (1/3)10^-3*10^6 or (1/3)(10^3)
+ (1/2)10^-2*10^4 or (1/2)(10^2)
+ 10^1*10^2 = 10^3
+ 10^2
= about 333+ 50 + 1000 + 100
or about 1483 I get
To convert the marginal cost function into a total cost function, you need to integrate the marginal cost function with respect to q. Here's how you can do it:
Part 1:
1. Identify the variables:
- r represents the marginal cost of widgets
- q represents the quantity of widgets produced
- TC represents the total cost of producing q widgets
2. Integrate the marginal cost function:
∫(dr/dq) dq = ∫(0.001Q^2 + 0.01Q + 10) dq
3. Integrate each term separately:
∫(0.001Q^2 + 0.01Q + 10) dq = (0.001/3)Q^3 + (0.01/2)Q^2 + 10Q + C1
Note: C1 represents the constant of integration. It is included to account for any fixed costs.
4. Add the total fixed costs:
TC = (0.001/3)Q^3 + (0.01/2)Q^2 + 10Q + C1 + 500
Since the total fixed costs are given as $500, you add it to the integrated equation.
Therefore, the total cost function is TC = (0.001/3)Q^3 + (0.01/2)Q^2 + 10Q + 500.
Part 2:
To determine total costs when Q = 100, simply substitute Q = 100 into the total cost function:
TC = (0.001/3)(100)^3 + (0.01/2)(100)^2 + 10(100) + 500
TC = 10 + 50 + 1000 + 500
TC = 1560
Therefore, the total costs when Q = 100 is $1560.