In the market for automobiles, assume that there is only one firm with 100% market power, and this firm produces only one kind of car. The market...

The key here is recognizing that we have a monopoly on our hands, and under a monopoly, the price will not be set equal to the marginal cost. Instead, marginal cost will be equal to marginal revenue, which is strictly lower than the price because increasing the number of goods sold requires decreasing the price.

So first, let's compute the marginal revenue. This is the derivative of revenue (P*Q) with respect to quantity (Q):

MR = d[P*Q]/dQ = dP/dQ * Q + P

We can substitute in our demand curve:
Q_d = 40,000 – (1/100)*P

P = 100*(40,000 - Q)

dP/dQ = -100

MR = (-100) * Q + 100*(40,000 - Q)

MR = 4,000,000 - 200*Q

Now we want to set this equal to marginal cost. Marginal cost is the derivative of total cost, again with respect to quantity.

For some reason they gave us quantity as N instead of Q, but it comes to the same thing:

TC = 25 * Q^2

MC = d[TC]/dQ = 50 * Q

Now the monopoly sets marginal revenue equal to marginal cost:

MR = MC

4,000,000 - 200*Q = 50*Q

4,000,000 = 250*Q

16,000 = Q (N automobiles)

Substitute this back into the demand curve:

P = 100*(40,000 - Q)

P = 100*(40,000 - 16,000)

P = $2,400,000

These numbers seem a bit weird ($2.4 million for each car? These must be Aston Martins!), but that's what they gave us. They do work out mathematically.

To get the profit made, we find the revenue P*Q and subtract the total cost:

profit = P*Q - TC

profit = P*Q - 25*Q^2

profit = (2,400,000)*(16,000) - 25*(16,000)^2 = 38,400,000,000  - 6,400,000,000

profit = $32,000,000,000

The profit is $32 billion. Not bad!