Question 1189635: The Métis buffalo hunts were important social events held twice a year and they were highly organized. In 1840, 1,630 people from the Métis of the Red River settlements headed south toward their traditional hunting prairie, which interestingly was approximately triangular- shaped, to take part on such a summer hunt. Let's consider the math of such a hunt through the lens of the hunter. --The hunter at position A is located 300m directly north of the basecamp, at position B. The hunter estimates the angle towards a distant bison herd located at position Cat approximately 60 between the herd and the camp. If the hunter is alerted that the bison look like they're about to move on and will probably only remain in their present position for another 2 minutes, how fast must the hunter travel in order to reach the bison in time? (answer in m/s)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
1. **Visualize the Triangle:** The hunter, basecamp, and bison form a triangle. The hunter is at point A, the basecamp is at point B, and the bison are at point C. We know the distance AB (300m) and the angle ACB (60°). We need to find the distance AC.
2. **Use the Law of Sines:** The Law of Sines is useful for solving triangles when you have an angle and its opposite side, and another angle or side. In this case:
AC / sin(B) = AB / sin(C)
We don't know angle B yet, but we do know that the sum of angles in a triangle is 180. So, if we let angle A be represented by x, then x + B + 60 = 180, which simplifies to B = 120 - x. Since we don't know angle A, we can't use the law of sines yet. However, since we know that the triangle is a right triangle, we can use trigonometry.
3. **Use Trigonometry:** Since the hunter is directly north of the basecamp, angle ABC is 90 degrees. This means we have a right triangle. We can use the tangent function to find the distance BC.
tan(60) = AB / BC
BC = AB / tan(60)
BC = 300 / √3
BC ≈ 173.21m
Now we can use the Pythagorean theorem to find AC
AC = √(AB^2 + BC^2)
AC = √(300^2 + 173.21^2)
AC = √(90000 + 30000)
AC = √(120000)
AC ≈ 346.41m
4. **Calculate Required Speed:** The hunter has 2 minutes (120 seconds) to reach the bison.
Speed = Distance / Time
Speed = 346.41 m / 120 s
Speed ≈ 2.89 m/s
**Answer:** The hunter must travel at approximately 2.89 m/s to reach the bison in time.
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