Question

Gadongville and Ledesma Subdivision are two villages located on opposite sides of an island. Hacienda Alcayde is located well away in the island and has a straight highway connection to each village. It is 84.6 miles from Gadongville to Hacienda Alcayde and 33.8 miles from Hacienda Alcayde to Ledesma Subdivision. A surveyor notes that the highway from Ledesma Subdivision to Hacienda Alcayde makes an angle measure of 63°20¢ with the line of sight from Ledesma Subdivision to Gadongville. How far apart are the two villages?

Answer 1

Answer on question 34177 – Math – Geometry

Gadongville and Ledesma Subdivision are two villages located on opposite sides of an island. Hacienda Alcayde is located well away in the island and has a straight highway connection to each village. It is 84.6 miles from Gadongville to Hacienda Alcayde and 33.8 miles from Hacienda Alcayde to Ledesma Subdivision. A surveyor notes that the highway from Ledesma Subdivision to Hacienda Alcayde makes an angle measure of $63{}^{\circ}20\text{C}$ with the line of sight from Ledesma Subdivision to Gadongville. How far apart are the two villages?

Solution

Using the law of cosine we get

$$84.6^{2} = 33.8^{2} + x^{2} - 2 * 33.8x \cos 63{}^{\circ}20$$ $$7157.16 = 1142.44 + x^{2} - 30.37x$$ $$x^{2} - 30.37x - 6014.72 = 0$$ $$D = 24058.88 + 92.34 = 24981.22$$ $$x_{1} = \frac{30.37 - 158.05}{2} < 0; \quad x_{2} = \frac{30.37 + 158.05}{2} = 94.21 \text{ feet}.$$

Answer: 94.21 feet.

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