Tc=100+55q-100q²+Q³ .find ATC And MTC
To find the Average Total Cost (ATC) and Marginal Total Cost (MTC), we need to differentiate the Total Cost (TC) equation with respect to quantity (q).
First, let's differentiate TC with respect to q to find the equation for Marginal Cost (MC):
MC = dTC / dq
To differentiate TC with respect to q, we need to differentiate each term individually.
The derivative of 100 with respect to q is 0 because it is a constant term.
The derivative of 55q with respect to q equals 55 because it is a linear term.
To differentiate -100q² with respect to q, we apply the power rule. The derivative of q^n with respect to q is nq^(n-1). So, the derivative of -100q² with respect to q is -200q.
To differentiate Q³ with respect to q, we apply the power rule again. The derivative of q^n with respect to q is nq^(n-1). So, the derivative of Q³ with respect to q is 3Q².
Now we can combine these derivatives to find the equation for MC:
MC = 55 - 200q + 3Q²
To find the Average Total Cost (ATC), we divide the Total Cost (TC) by the quantity (q):
ATC = TC / q
Substituting the given TC equation into the ATC equation:
ATC = (100+55q-100q²+Q³) / q
To simplify this expression, we will divide each term by q:
ATC = 100/q + 55 - 100q + Q²
So, the equation for the Average Total Cost (ATC) is 100/q + 55 - 100q + Q².
Finally, the Marginal Total Cost (MTC) is equal to the Marginal Cost (MC).
Therefore, the equations for the Average Total Cost (ATC) and Marginal Total Cost (MTC) are:
ATC = 100/q + 55 - 100q + Q²
MTC = 55 - 200q + 3Q²