A triangular plot of land is enclosed by a fence.  One side of the fence is 30.3 m long and an adjacent side of the fence is 18.5 m long.  The inclusive angle between the two sides is 61 degrees.  Determine the third side of the fence to the nearest tenth of a metre.

To determine the third side of the fence, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\) and angle \(C\) opposite side \(c\):

\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

In this problem, we are given:
- \(a = 30.3\) m
- \(b = 18.5\) m
- \(C = 61^\circ\)

We need to find the length of side \(c\). Plugging the given values into the Law of Cosines formula:

\[ c^2 = 30.3^2 + 18.5^2 - 2 \cdot 30.3 \cdot 18.5 \cdot \cos(61^\circ) \]

First, calculate each term:

\[ 30.3^2 = 918.09 \]
\[ 18.5^2 = 342.25 \]
\[ 2 \cdot 30.3 \cdot 18.5 = 1119.3 \]
\[ \cos(61^\circ) \approx 0.4848 \]

Now, substitute these values into the equation:

\[ c^2 = 918.09 + 342.25 - 1119.3 \cdot 0.4848 \]
\[ c^2 = 1260.34 - 542.53 \]
\[ c^2 = 717.81 \]

Finally, take the square root to find \(c\):

\[ c = \sqrt{717.81} \approx 26.8 \]

Therefore, the length of the third side of the fence is approximately 26.8 meters.

Would you like more details or have any questions?

Here are 5 more questions you can ask:
1. How do you use the Law of Sines to solve for unknown angles in a triangle?
2. Can you explain how to find the area of a triangle given its sides using Heron's formula?
3. What is the process for solving a quadratic equation by factoring?
4. How do you determine the derivative of a polynomial function?
5. Can you explain how to integrate a simple trigonometric function?

Tip: When using trigonometric functions on a calculator, ensure it is set to the correct mode (degrees or radians) based on the problem's requirements.

Learn how to calculate the length of the third side of a triangle using the Law of Cosines with a detailed step-by-step solution. Given sides and angle details are provided. Suitable for high school students in Grades 10-12.

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