To determine the perimeter of the pool, we can use the information provided in the problem to set up an equation.

Step 1: Define the variables

• Let $L$ represent the length of the pool (in feet).
• Let $W$ represent the width of the pool (in feet).

Step 2: Use the perimeter formula

The perimeter $P$ of a rectangle is given by: $P = 2L + 2W$

Given that the perimeter of the pool is 406 feet, we have: $2L + 2W = 406$

Step 3: Incorporate the relationship between length and width

We are also told that the length $L$ is 15 feet more than three times the width $W$. This can be expressed as: $L = 3W + 15$

Step 4: Substitute the length equation into the perimeter equation

Substitute $L = 3W + 15$ into the perimeter equation: $2(3W + 15) + 2W = 406$

Final Equation

The equation that could be used to determine the perimeter of the pool is: $2(3W + 15) + 2W = 406$

This equation can be simplified and solved to find the width $W$ of the pool.

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

Here are some related questions you might consider:

1. How would you solve the equation $2(3W + 15) + 2W = 406$ to find $W$?
2. What is the perimeter equation if the length is four times the width?
3. How does the perimeter formula change if the shape is a square instead of a rectangle?
4. Can you write the equation for perimeter if both length and width are unknown but in terms of their ratio?
5. How would the equation change if the length was instead 20 feet more than three times the width?

Tip: When writing equations from word problems, clearly define each variable and ensure that all relationships in the problem are represented in the equation.