To solve for the length $b$ in the given triangle, we can apply the Law of Sines. The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Where:

• $a = 27.2 \, \text{cm}$
• $A = 43^\circ$
• $B = 72^\circ$
• $C = 180^\circ - (43^\circ + 72^\circ) = 65^\circ$

Now we can use the ratio:

$\frac{b}{\sin B} = \frac{a}{\sin A}$

Substitute the known values:

$\frac{b}{\sin 72^\circ} = \frac{27.2}{\sin 43^\circ}$

Solve for $b$:

$b = \frac{27.2 \times \sin 72^\circ}{\sin 43^\circ}$

Using approximate values for the sine functions:

$b = \frac{27.2 \times 0.9511}{0.6820} \approx \frac{25.874}{0.6820} \approx 37.93$

To the nearest tenth of a centimeter, $b \approx 37.9 \, \text{cm}$.

Would you like more details on this process or help with further steps?

Here are five related questions:

1. What is the Law of Cosines and when would you use it?
2. How would you calculate the length of side $c$ in this triangle?
3. What are the applications of the Law of Sines in real-world problems?
4. How do you determine if a triangle is obtuse or acute?
5. Why is it important to know the sum of angles in a triangle?

Tip: Always ensure angle measurements are in degrees or radians as needed for trigonometric calculations.