To approximate the length of a lake, surveryor starts at one end of the lake and walks 156 yards. He then turns 110° and walk 198 yards until he arrives at the other end of the lake. Approximately how long is the lake? (round to nearest yard)
We can use the Law of Cosines to approximate the length of the lake. Here's how:
Let's call the distance we're trying to find "x". We know two sides and the included angle of the triangle formed by the starting point, ending point, and the point where the surveyor turned. Using the Law of Cosines, we have:
x^2 = 156^2 + 198^2 - 2(156)(198)cos(110°)
x^2 = 24336 + 39204 - (61848)(-0.34202)
x^2 = 63596.79915
x ≈ 252.16 yards
Therefore, the lake is approximately 252 yards long.
To solve this problem, we can use the concept of trigonometry and the Law of Cosines.
First, let's draw a diagram to visualize the scenario:
```
x
-----
/ \
/ \
(y)/ \ (w)
/ \
---------------
```
In the diagram, we have a right triangle with sides x, y, and w.
We are given the following information:
- The surveyor walks 156 yards, which is the side y.
- The surveyor turns 110° and walks 198 yards, which is the side w.
To find the length of the lake (x), we need to use the Law of Cosines, which states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case:
- a = y = 156 yards
- b = w = 198 yards
- C = 110°
Substituting these values into the formula:
x^2 = 156^2 + 198^2 - 2 * 156 * 198 * cos(110°)
Let's calculate the value of x:
x^2 ≈ 24,336
Taking the square root of both sides:
x ≈ 156.15 yards
Therefore, the approximate length of the lake is 156.15 yards (rounded to the nearest yard).
To approximate the length of the lake, we can use the concept of the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we can consider the starting point of the surveyor, the ending point of the surveyor, and the far end of the lake as the vertices of a triangle.
Let's label the starting point as point A, the ending point as point B, and the far end of the lake as point C. We know the lengths of two sides, AB = 156 yards and BC = 198 yards, and the angle between these sides, angle BAC = 110°.
Let's use the Law of Cosines formula:
c^2 = a^2 + b^2 - 2 * a * b * cos(C)
where c is the length of side AC, a is the length of side AB, b is the length of side BC, and C is the angle opposite side c.
Substituting the known values:
AC^2 = 156^2 + 198^2 - 2 * 156 * 198 * cos(110°)
To calculate the approximate length of the lake, we need to find the square root of AC^2:
AC ≈ √(156^2 + 198^2 - 2 * 156 * 198 * cos(110°))
Using a calculator to perform the calculations:
AC ≈ √(24336 + 39204 - 61584 * cos(110°))
AC ≈ √(24336 + 39204 - 61584 * -0.34202014)
AC ≈ √(24336 + 39204 + 21103.9349)
AC ≈ √(84643.9349)
AC ≈ 290.9994 yards
Rounding to the nearest yard, the approximate length of the lake is 291 yards.