Final answer:
By applying the Law of Cosines to the triangle formed by you, the bear, and your grandmother, and using the distances and angle given, we find that the bear is approximately 21.3 feet away from your grandmother.
Step-by-step explanation:
To find out how far the bear is from your grandmother, we need to apply trigonometry to a scenario where you're at one vertex of a triangle, the bear at another, and your grandmother at the third vertex.
When you turn 35 degrees to your left, this angle becomes one of the angles in our triangle. Using the information given:
- You are 37 feet from the bear.
- You are 29 feet from your grandmother.
- The angle between the two lines of sight is 35 degrees.
We need to use the Law of Cosines to find the distance between the bear and your grandmother.
The Law of Cosines is stated as c^2 = a^2 + b^2 - 2ab*cos(C), where a and b are the sides adjacent to angle C, and c is the side opposite angle C.
Substituting our values into the formula gives us:
c^2 = 37^2 + 29^2 - 2*37*29*cos(35 degrees)
Let's calculate that:
c^2 = 1369 + 841 - 2*37*29*cos(35 degrees)
c^2 = 2210 - 2146*cos(35 degrees)
To find cos(35 degrees), we can use a calculator:
cos(35 degrees) ≈ 0.8192
Now plug this back into the equation:
c^2 = 2210 - 2146*0.8192
c^2 = 2210 - 1758.5072
c^2 = 451.4928
c ≈ √451.4928
c ≈ 21.25
Therefore, the bear is approximately 21.3 feet away from your grandmother, rounding to the nearest tenth of a foot.