The problem describes a monopolist's total cost function $TC = 0.1Q^2 - 2Q + 100$ and marginal cost function $MC = 0.2Q - 2$. The market demand is given by $Q = 86 - P$, and the marginal revenue is $MR = 86 - 2Q$. We need to find the profit-maximizing output.

1. Problem Description

The problem describes a monopolist's total cost function $TC = 0.1Q^2 - 2Q + 100$ and marginal cost function $MC = 0.2Q - 2$. The market demand is given by $Q = 86 - P$, and the marginal revenue is $MR = 86 - 2Q$. We need to find the profit-maximizing output.

2. Solution Steps

To find the profit-maximizing output, we need to set marginal revenue (MR) equal to marginal cost (MC):

$MR = MC$

We are given:

$MR = 86 - 2Q$

$MC = 0.2Q - 2$

Setting them equal:

$86 - 2Q = 0.2Q - 2$

Now, we solve for Q:

$86 + 2 = 0.2Q + 2Q$

$88 = 2.2Q$

$Q = \frac{88}{2.2}$

$Q = 40$

3. Final Answer

The profit-maximizing output is

4

0.