Answer
The measure of the angle where the ravines meet is about $100.3^{\circ}$.
Work Step by Step
First, we want to know what side is opposite the angle in question. The side that is opposite to the angle we are looking for, $x$, is the side that measures $20$ ft., so let's plug in what we know into the formula for the law of cosines:
$20^2 = 12^2 + 14^2 - 2(12)(14)$ cos $x$
Evaluate exponents first, according to order of operations:
$20^2 = 144 + 169 - 2(12)(14)$ cos $x$
Add to simplify on the right side of the equation:
$400 = 340 - 2(12)(14)$ cos $x$
Multiply on the right side of the equation:
$400 = 340 - 336$ cos $x$
Subtract $340$ from each side of the equation to move constants to the left side of the equation:
$60 = -336$ cos $x$
Divide each side by $-336$:
cos $\angle x = \frac{60}{-336}$
Take $cos^{-1}$ to solve for $\angle x$:
$m \angle x \approx 100.3^{\circ}$
The measure of the angle where the ravines meet is about $100.3^{\circ}$.
