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To find the length of side \( AC \) in the triangle, we can use the **Law of Cosines**. This law is useful for solving triangles when we know two sides and the included angle between them.

### Law of Cosines Formula
For a triangle with sides \( a \), \( b \), and \( c \), and angle \( \theta \) opposite side \( c \), the Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]

### Applying the Law of Cosines
In this case:
- \( AB = 36.19 \, \text{ft} \) (side \( a \))
- \( CB = 18.26 \, \text{ft} \) (side \( b \))
- \( \angle ABC = 73^\circ \) (included angle)

We want to find \( AC \) (side \( c \)).

1. Substitute into the Law of Cosines:
   \[
   AC^2 = (36.19)^2 + (18.26)^2 - 2 \cdot 36.19 \cdot 18.26 \cdot \cos(73^\circ)
   \]

2. Calculate each part:
   - \( (36.19)^2 = 1309.92 \)
   - \( (18.26)^2 = 333.38 \)
   - \( 2 \cdot 36.19 \cdot 18.26 = 1324.27 \)
   - \( \cos(73^\circ) \approx 0.2924 \)
   
3. Substitute and simplify:
   \[
   AC^2 = 1309.92 + 333.38 - 1324.27 \cdot 0.2924
   \]
   \[
   AC^2 = 1309.92 + 333.38 - 387.23
   \]
   \[
   AC^2 = 1256.07
   \]

4. Take the square root to find \( AC \):
   \[
   AC \approx \sqrt{1256.07} \approx 35.45 \, \text{ft}
   \]

### Answer
The length of side \( AC \) is approximately **35.45 feet**, rounded to the nearest hundredth.

Would you like further details or have any questions?

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### Additional Practice Questions:
1. How would the answer change if the angle \( \angle ABC \) was larger than \( 73^\circ \)?
2. What if \( AB \) was shorter than \( CB \); how would that impact the length of \( AC \)?
3. If the included angle was \( 90^\circ \), what would the calculation look like?
4. How could you use the Law of Cosines to solve for the angle if all three side lengths were known?
5. Can you verify the result using the Law of Sines?

### Tip:
The Law of Cosines is a generalization of the Pythagorean theorem, which applies to all triangles, not just right triangles.

A professor at a school of Architecture designed an asymmetrical A-frame house pictured below. The included angle between sides AB and CB, which measure 36.19 ft and 18.26 ft, respectively, measures 73°. Find the length of side AC.

Learn how to use the Law of Cosines to find the length of side AC in a triangle with sides AB = 36.19 ft, CB = 18.26 ft, and an included angle of 73°. This trigonometry problem is ideal for students in Grades 10-12.

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