Set up the inequality.

Let $B_0$ be the initial bank balance, $W$ be the weekly withdrawal amount, and $n$ be the number of weeks. The balance after $n$ weeks is $B_0 - n \times W$. Anna wants to maintain a balance of at least Tshs 200,000.

$$B_0 - n \times W \ge 200,000$$

Given:
$B_0 = \text{Tshs } 500,000$
$W = \text{Tshs } 40,000$

Substitute the values into the inequality:

$$500,000 - n \times 40,000 \ge 200,000$$

Subtract 500,000 from both sides:

$$-n \times 40,000 \ge 200,000 - 500,000$$ $$-n \times 40,000 \ge -300,000$$

Divide both sides by -40,000. Remember to reverse the inequality sign when dividing by a negative number:

$$n \le \frac{-300,000}{-40,000}$$ $$n \le \frac{30}{4}$$ $$n \le 7.5$$

Since $n$ represents the number of weeks, it must be a whole number. Anna can withdraw money for a maximum of 7 full weeks while maintaining a balance of at least Tshs 200,000. If she withdraws for 8 weeks, her balance will fall below Tshs 200,000.

The maximum number of weeks she can withdraw money is $\text{7 weeks}$.