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To determine the length of \( CB \) in the triangle \( \triangle ABC \) with the given information:

1. Side \( AC = 32 \) meters
2. Side \( AB = 40 \) meters
3. Angle \( \angle CAB = 58^\circ \)

### Step-by-Step Solution Using the Cosine Law

The Cosine Law formula is:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where:
- \( c \) is the side opposite angle \( C \),
- \( a \) and \( b \) are the other two sides of the triangle, and
- \( \cos(C) \) is the cosine of the included angle.

In this triangle:
- \( c = CB \) (the unknown we need to find),
- \( a = AC = 32 \) meters,
- \( b = AB = 40 \) meters, and
- \( C = 58^\circ \).

### Applying the Values

\[
CB^2 = 32^2 + 40^2 - 2 \cdot 32 \cdot 40 \cdot \cos(58^\circ)
\]

Calculating each term:
1. \( 32^2 = 1024 \)
2. \( 40^2 = 1600 \)
3. \( 2 \cdot 32 \cdot 40 = 2560 \)
4. Calculate \( \cos(58^\circ) \approx 0.5299 \).

So,
\[
CB^2 = 1024 + 1600 - 2560 \cdot 0.5299
\]
\[
CB^2 = 2624 - 1356.54
\]
\[
CB^2 \approx 1267.46
\]
\[
CB \approx \sqrt{1267.46} \approx 35.6 \text{ meters}
\]

### Answer
The length of \( CB \) is approximately **35.6 meters**.

Would you like a further breakdown of any part of this calculation? 

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### Additional Practice Questions

1. How would the length of \( CB \) change if \( \angle CAB \) were 45 degrees instead?
2. Can you solve for \( \angle ABC \) given the side lengths now?
3. What is the area of \( \triangle ABC \) using the side lengths found?
4. How would you apply the Sine Rule for this triangle if we needed \( \angle ABC \)?
5. Can you find \( \cos(CAB) \) in radians instead of degrees?

### Tip
In trigonometry problems involving triangles, always check if the Cosine Law or Sine Law is more suitable based on the given data.

Determine the length of CB to the nearest tenth of a meter in triangle ABC where side AC is 32 m, side AB is 40 m, and angle CAB is 58 degrees.

Solve for the length of CB in triangle ABC using the Cosine Law. Given AC = 32 m, AB = 40 m, and angle CAB = 58 degrees, find the length of CB to the nearest tenth of a meter. This solution covers key trigonometry concepts suitable for Grades 10-12.

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