triangle side length using law of cosines

1. Identify the problem type: This is a geometry problem that can be solved using the Law of Cosines. We have a triangle formed by the boat and the two docks, where we know two side lengths and the included angle.

1. Define the variables: Let $a = 500$ feet, $b = 600$ feet, and the angle between them $\theta = 45^{\circ}$. We want to find the distance $c$ between the two docks.

3. Apply the Law of Cosines: The Law of Cosines states that $c^2 = a^2 + b^2 - 2ab \cos(\theta)$. Substituting our values:
$$c^2 = 500^2 + 600^2 - 2(500)(600) \cos(45^{\circ})$$

4. Calculate the values:
$$c^2 = 250,000 + 360,000 - 600,000 \cdot \frac{\sqrt{2}}{2}$$
$$c^2 = 610,000 - 300,000 \cdot \sqrt{2}$$
Using $\sqrt{2} \approx 1.4142$:
$$c^2 \approx 610,000 - 300,000(1.4142)$$
$$c^2 \approx 610,000 - 424,264$$
$$c^2 \approx 185,736$$

5. Solve for c:
$$c = \sqrt{185,736} \approx 430.97$$
Rounding to the nearest whole number, we get $431$ feet.