Question

Primary school students are playing hide and seek at recess as the counting begins Jon runs 100 feet east , turns 45 degrees north of east and runs another 120 feet to find a hiding spot . Lisa finds her hiding spot by running 120 feet west , turning 32 degrees south of west , and continuing another 100 feet which students hiding spot is closer to starting point ?

Answer 1

Jon's hiding spot is closer to the starting point.

Explanation:

Jon's final position forms a triangle with sides 100 ft and 120 ft respectively with an angle of 180 - 45 = 135 between them. Since the angle between the 100 ft direct east and 120 ft direction is 45, since Jon turns 45 after moving for 100 ft and then moves another 120 ft to his final destination.

Also, Lisa's final position forms a triangle with sides 120 ft and 100 ft respectively with an angle of 180 - 32 = 148 between them. Since the angle between the 120 ft direction west and 100 ft direction south of west is 45, since Lisa turns 32 after moving for 120 ft and then moves another 100 ft to her final destination.

Using the cosine rule,

a² = b² + c² - 2abcosФ, we find the length of the third side of each triangle which is their distance from the starting point.

So, for Jon

a² = b² + c² - 2abcosФ where b = 100 ft, c = 120 ft and Ф = 135°

Thus,

a² = (100 ft)² + (120 ft)² - 2(100 ft)(120 ft)cos135°

a² = 10000 ft² + 14400 ft² - 24000 ft²cos135°

a² = 10000 ft² + 14400 ft² - 24000 ft²(-0.7071)

a² = 24400 ft² + 16970.56 ft²

a² = 41370.56 ft²

a = √(41370.56 ft²)

a = 203.4 ft

for Lisa

a'² = b'² + c'² - 2a'b'cosФ' where b = 120 ft, c = 100 ft and Ф = 148°

Thus,

a'² = (120 ft)² + (100 ft)² - 2(120 ft)(100 ft)cos148°

a'² = 14400 ft² + 10000 ft² - 24000 ft²cos148°

a'² = 14400 ft² + 10000 ft² - 24000 ft²(-0.8481)

a'² = 24400 ft² + 20353.15 ft²

a'² = 44753.15 ft²

a' = √(44753.15 ft²)

a' = 211.56 ft

a' ≅ 211.6 ft

Since a = 203.4 ft < a' = 211.6 ft,

Jon's hiding spot is closer to the starting point.