I have an exam in two days. I need an explanation to this question please.
MC = 10+5Q
derive an equation for total cost
Integrate the MC function. MC is the first derivitive of TC. So,
TC = 10Q + (5/2)Q^2 + C
A television station is considering the sale of promotional DVDs. It can have the DVDs produced by one of two suppliers. Supplier A will charge the station a set-up fee of $1200 plus $2 for each DVD; supplier B has no set-up fee and will charge $4 per DVD. The station estimates its demand for the DVDs to be given by Q = 1,600 - 200P, where P is the price in dollars and Q is the number of DVDs. The price equation is P = 8 -Q/200
To derive an equation for total cost from the given information, we need to understand that total cost (TC) is the sum of two components: the fixed cost (FC) and the variable cost (VC).
In the provided equation, MC (marginal cost) represents the rate of change of total cost with respect to the quantity produced (Q). It is given by the equation:
MC = dTC/dQ
Since MC represents the derivative of TC with respect to Q, we can determine TC by integrating MC with respect to Q. Integrating MC will give us the total cost function.
In this case, MC = 10 + 5Q.
To integrate this equation, we need to keep in mind that the integration of Q^n with respect to Q is (1/n+1)*Q^(n+1).
So, integrating MC = 10 + 5Q with respect to Q:
∫MC dQ = ∫(10 + 5Q) dQ
The integral of MC with respect to Q is calculated as follows:
∫MC dQ = ∫(10 + 5Q) dQ
= 10Q + (5Q^2)/2 + C
Here, C represents the constant of integration. Since we are interested in deriving the total cost equation, we can assume that the constant of integration already incorporates the fixed cost (FC), which is constant regardless of the quantity produced.
Therefore, the equation for total cost (TC) can be obtained by replacing the integral results with TC:
TC = 10Q + (5Q^2)/2 + C
This equation represents the total cost as a function of the quantity produced (Q), with the fixed cost (FC) included in the constant term (C).