The equation given is:

$L + 1.25 \cdot \sqrt{S} - 9.8 \cdot D^{\frac{1}{3}} = 16.296$

You need to find $S$ when the length $L = 20.52 \, \text{m}$ and the displacement $D = 17.72 \, \text{m}^3$.

Let's substitute the known values of $L$ and $D$ into the equation and solve for $S$.

$20.52 + 1.25 \cdot \sqrt{S} - 9.8 \cdot (17.72)^{\frac{1}{3}} = 16.296$

First, calculate $9.8 \cdot (17.72)^{\frac{1}{3}}$, and then solve for $S$. I'll handle the calculations step by step.The sail area $S$ for the boat with a length of 20.52 meters and a displacement of 17.72 cubic meters is approximately $291.05 \, \text{m}^2$.

Would you like more details on the steps, or do you have any questions?

Here are some related questions to expand your understanding:

1. How does the displacement $D$ affect the sail area in the equation?
2. What happens to the sail area $S$ if the length of the boat is increased?
3. Can you solve for $L$ if the sail area $S$ and displacement $D$ are known?
4. How would the formula change if a different coefficient is used for $S^{1/2}$?
5. What is the physical meaning of the cube root of displacement $D$?

Tip: When solving for variables under a square root, it's important to isolate the square root term before squaring both sides to avoid errors.