To determine the distance to an oil platform in the Pacific Ocean, from both ends of a beach, a surveyor measures the angle to the platform from each end of the beach. The angle made with the shoreline from one end of the beach is 83 degrees, from the other end 78.6 degrees. If the beach is 950 yards long, what are the distances to the oil platform from both ends of the beach? Please help if you can and show all the work so I will learn. Thanks.
To determine the distances to the oil platform from both ends of the beach, we can use trigonometry and the given angles.
Let's denote the distance from one end of the beach to the oil platform as \(x\), and from the other end as \(y\).
Now, we can use the tangent function to relate the angles to the distances:
For the first angle of 83 degrees, we have:
\(\tan(83^\circ) = \frac{x}{950}\)
For the second angle of 78.6 degrees, we have:
\(\tan(78.6^\circ) = \frac{y}{950}\)
To find \(x\) and \(y\), we can solve these two equations.
First, let's solve for \(x\):
\(\tan(83^\circ) = \frac{x}{950}\)
Rearranging the equation:
\(x = \tan(83^\circ) \times 950\)
Using a calculator, we find:
\(x \approx 3735.69\) yards (rounded to two decimal places)
Now, let's solve for \(y\):
\(\tan(78.6^\circ) = \frac{y}{950}\)
Rearranging the equation:
\(y = \tan(78.6^\circ) \times 950\)
Using a calculator, we find:
\(y \approx 3478.41\) yards (rounded to two decimal places)
So, the distances to the oil platform from one end of the beach are approximately 3735.69 yards, and from the other end is approximately 3478.41 yards.