Erik and Zach are camping. Erik leaves Zach at the campsite and walks 3.6 miles. He then turns at a 114˚ angle and walks another 2.3 miles. If Erik were to walk directly back to Zach, how far would he walk? (draw a picture to help you solve)
did you draw a diagram? If so, you can see that the distance x is found using the law of cosines:
x^2 = 3.6^2 + 2.3^2 - 2*3.6*2.3*cos66˚
To solve this problem, we need to visualize the scenario and use basic geometry.
First, let's draw a diagram to represent the given information.
```
3.6 miles
---------------
| |
| |
| |
| Erik |
| |
| |
---------------
^
| 2.3 miles
```
In this diagram, Erik starts at the campsite and walks 3.6 miles. Then he turns at a 114˚ angle and walks another 2.3 miles. We are asked to find how far Erik would walk if he were to go directly back to the starting point.
To solve this, we can use the Law of Cosines. The Law of Cosines states that, in a triangle, if we know the lengths of two sides and the angle between them, we can find the length of the remaining side using the formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c represents the side we want to find (the distance Erik walks directly back to Zach), a and b are the known sides in the triangle, and C is the angle between the known sides.
In our case, the known sides are 3.6 miles and 2.3 miles, and the angle C is 114˚. We want to find the side c.
Plugging the values into the formula, we have:
c^2 = (3.6)^2 + (2.3)^2 - 2 * 3.6 * 2.3 * cos(114˚)
Now, we can use a calculator to evaluate this expression:
c^2 ≈ 12.96 + 5.29 - 16.56 * (-0.414)
c^2 ≈ 12.96 + 5.29 + 6.85
c^2 ≈ 25.1
Taking the square root of both sides, we find:
c ≈ √25.1
c ≈ 5.01 miles
Therefore, Erik would need to walk approximately 5.01 miles to go directly back to the campsite where Zach is located.