Let's start by solving part (a) and then work on part (b).

Part (a) — Solving for $K$ in the formula:

The given formula is:

$F = \frac{9}{5} (K - 273.15) + 32$

To solve for $K$:

1. Subtract 32 from both sides:

$F - 32 = \frac{9}{5} (K - 273.15)$

2. Multiply both sides by $\frac{5}{9}$:

$\frac{5}{9} (F - 32) = K - 273.15$

3. Add 273.15 to both sides:

$K = \frac{5}{9} (F - 32) + 273.15$

So, the formula solved for $K$ is:

$K = \frac{5}{9} (F - 32) + 273.15$

Part (b) — Converting 180°F to Kelvin:

Now, using the formula we derived for $K$, plug in $F = 180$:

$K = \frac{5}{9} (180 - 32) + 273.15$

I'll calculate this next.The temperature 180°F is approximately $355.37 \, K$ when rounded to the nearest hundredth.

Do you have any further questions or would you like more details on this process?

Here are 5 related questions you can explore:

1. How do you convert from Kelvin to Celsius?
2. What is the Kelvin equivalent of the freezing point of water in Fahrenheit?
3. How does the Kelvin scale differ from the Celsius and Fahrenheit scales?
4. How do you convert Celsius to Fahrenheit?
5. What real-world scenarios require the use of the Kelvin scale?

Tip: The Kelvin scale is commonly used in scientific fields, especially in thermodynamics, because it starts at absolute zero, where molecular motion stops.