Suppose the demand curve facing a monopolist is estimated to be:  P
= 140 –3Q.
     The firm’s marginal cost is represented by:     MC = 4 + 2Q.  

     a.  Determine the monopolist’s profit-maximizing level of output,
and what price will the monopolist charge?

Hello.

A monopolist's profit-maximizing output is reached when marginal cost
(MC) is equal to marginal revenue (MR).

You already have the function for marginal cost ( MC = 4 + 2Q).

To solve the problem, you just need to calculate marginal revenue
(MR).

To get marginal revenue, first start with the function for Total
Revenue (TR).

TR = P * Q

Since your demand curve indicates that P = 140 – 3Q , you can
subsitute that in for P in the TR function:

TR = P * Q 
TR = (140 - 3Q) * Q
TR = 140Q - 3Q^2

You calculate marginal revenue by taking the first derivative of the
TR function:
MR = dTR/dQ = 140 - 6Q

Now, we go back to the principle that the monopolist maximizes profits
when marginal cost (MC) is equal to marginal revenue (MR).

MC = 4 + 2Q
MR = 140 - 6Q

Thus, when MC = MR :
4 + 2Q = 140 - 6Q

Rearranging algebraically, we see that:
8Q = 136 
Q = 17

Now that we know that Q = 17, we just plug that back into the demand
function to get price:
P = 140 - 3Q = 140 - (3 * 17) = 89

Thus, the monopolist's profit-maximizing level of output is 17.
The monopolist will charge a price of 89.


For similar problems, see the following example:
#4 of Harvard Econ 1010a Problem Set, cached by Google:
http://216.239.51.100/search?q=cache:I0z75IBof9wC:icg.harvard.edu/~ec1010a/Problem_Sets/Ec1010a.ps4a.pdf+%22marginal+revenue%22+derivative+%22total+revenue%22+p+q&hl=en&ie=UTF-8&client=googlet
"Profit Maximization" example in University of North Carolina: Econ
100, cached by Google:
http://216.239.51.100/search?q=cache:UoIQ68GD8IoC:www.unc.edu/courses/2002fall/econ/100/001/10a101.pdf+mr+mc+%22marginal+cost%22+demand+slope&hl=en&ie=UTF-8&client=googlet

search strategy: mc, "marginal cost", demand, slope, p, q

I hope this helps.