1. Points $A$ and $B$ are separated by a lake. To find the distance between them, a surveyor locates a point $C$ on land such that $\angle C A B=48.6^{\circ}$. He also measures $C A$ as $312 \mathrm{ft}$ and $C B$ as $527 \mathrm{ft}$. Find the distance between $A$ and $B$.

Step 1 ：Given points $A$ and $B$ are separated by a lake. A surveyor locates a point $C$ on land such that $\angle C A B=48.6^\circ$. He also measures $C A$ as $312 \mathrm{ft}$ and $C B$ as $527 \mathrm{ft}$. We are asked to find the distance between $A$ and $B$.

Step 2 ：We can use the Law of Cosines to solve this problem. The Law of Cosines states that for any triangle with sides of lengths a, b, and c and an angle $\gamma$ opposite side c, the following equation holds: $c^2 = a^2 + b^2 - 2ab\cos(\gamma)$.

Step 3 ：In this case, we know the lengths of sides $CA$ and $CB$ and the measure of $\angle CAB$, so we can use these values to find the length of $AB$.

Step 4 ：Substitute the given values into the Law of Cosines: $AB^2 = CA^2 + CB^2 - 2*CA*CB*\cos(\angle CAB)$

Step 5 ：Substitute $CA = 312$, $CB = 527$, and $\angle CAB = 48.6^\circ$ into the equation.

Step 6 ：Calculate the value of $AB$ to get $AB \approx 396.99107762271905$

Step 7 ：Round the result to the nearest whole number to get $AB \approx 397$

Step 8 ：Final Answer: The distance between points $A$ and $B$ is approximately \(\boxed{397}\) feet.