Question

Alice and her friend Emma leave on different flights from the same airport. Alice's flight flies 100 miles due south, then turns 70° toward west and flies 50 miles. Emma's flight flies 100 miles due north, then turns 50° toward east and flies 50 miles. Which fight among you is farther from the airport? Explan your reasoning.

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Answer 1

Answer:

This question is based on the Law of Cosines.

Law of Cosines is a formula or a method in which we find the third side of a triangle given the other two sides along with the measure of the included angle.

The formula for calculating the third side is given as:

$\boxed{ \bf{c = \sqrt{a^2 + b^2 - 2.a.b.cos\:(\theta)}}}$

where,

a, b, c are sides of the triangle and Cos Ф is the measure of Cosine of the angle between them in degrees.

According to the question,

Alice's flight travels 100 miles towards south and takes a 70 degree turn to travel 50 miles due west. Interpreting this question as a diagram we get:

• a = 100 miles
• b = 50 miles
• Ф = 70°

Substituting in the formula we get:

$\implies c = \sqrt{ 100^2 + 50^2 - 2(100)(50) \times Cos(70)}\\\\\\\implies c = \sqrt{ 10000 + 2500 - 10000(0.342)}\\\\\\\implies c = \sqrt{12500 - 3420}\\\\\\\implies c = \sqrt{9080}\\\\\\\impliex \boxed{ \bf{ c \approx 95.28 \:miles}}$

Calculating the distance traveled by Emma, we get:

• a = 100 miles
• b = 50 miles
• Ф = 50°

$\implies c = \sqrt{100^2 + 50^2 - 2(100)(50) \times Cos\:(50)}\\\\\\\implies c = \sqrt{10000 + 2500 - 10000(0.642)}\\\\\\\implies c = \sqrt{12500 - 6420}\\\\\\\implies c = \sqrt{6080}\\\\\\\implies \boxed{ \bf{ c \approx 77.92 \:miles}}$

From the above results we can say that,

Alice's flight is more far from the airport than Emma's flight.