Question

the figure shows a 1125-yard-long sand beach and an oil platform in the ocean. the angle made with the platform from one end of the beach is \\(81^{\circ}\\) and from the other end is \\(76^{\circ}\\). find the distance of the oil platform, to the nearest tenth of a yard, from each end of the beach.

the platform is about yards from one end of the beach and yards from the other.
(use descending order. round to the nearest tenth as needed.)

Explanation:

Find the third angle of the triangle

The sum of angles in a triangle is \(180^\circ\).
Let the angle at the oil platform be \(C\).

$$ C = 180^\circ - 81^\circ - 76^\circ = 23^\circ $$

Apply the Law of Sines

Let \(a\) be the distance from the end with the \(76^\circ\) angle to the platform.
Let \(b\) be the distance from the end with the \(81^\circ\) angle to the platform.
Let \(c = 1125\text{ yards}\) be the length of the beach.

$$ \frac{a}{\sin(81^\circ)} = \frac{b}{\sin(76^\circ)} = \frac{1125}{\sin(23^\circ)} $$

Calculate the first distance

Solve for \(a\), the distance opposite the \(81^\circ\) angle:

$$ a = \frac{1125 \cdot \sin(81^\circ)}{\sin(23^\circ)} \approx \frac{1125 \cdot 0.98769}{0.39073} \approx 2844.2\text{ yards} $$

Calculate the second distance

Solve for \(b\), the distance opposite the \(76^\circ\) angle:

$$ b = \frac{1125 \cdot \sin(76^\circ)}{\sin(23^\circ)} \approx \frac{1125 \cdot 0.97030}{0.39073} \approx 2794.1\text{ yards} $$

Arrange in descending order

Comparing the two calculated distances:

$$ 2844.2\text{ yards} > 2794.1\text{ yards} $$

Answer:

The platform is about <blank>2844.2</blank> yards from one end of the beach and <blank>2794.1</blank> yards from the other.
(Use descending order. Round to the nearest tenth as needed.)