Solving Pool Dimensions Based on Surface Area

When designing or analyzing a rectangular pool, understanding the relationship between its dimensions and surface area is crucial. In this article, we will explore a specific problem where the width of a rectangular pool is three times its depth, and the length is twice the width. Given the internal surface area of the pool, we will derive the depth of the pool step by step.

Problem Statement

The width of a rectangular pool is three times its depth. The length is twice the width. If the internal surface area of the pool is 158.76 m2, what is the depth of the pool?

Given Information

Dimension Expression in Terms of Depth (x) Depth (d) x Width (w) 3x Length (l) 2w 6x

Internal Surface Area Calculation

To understand the internal surface area, we need to consider the bottom, the sides, and the ends of the pool, excluding the top surface. The internal surface area can be broken down as follows:

Bottom area: The area of the bottom of the pool is given by the length times the width, which is (6x times 3x 18x^2). Side area: The area of the sides is the perimeter of the bottom times the depth. The perimeter is (2 times (6x 3x) 18x), so the side area is (18x times x 18x^2). End area: The area of the ends is the width times the depth, which is (3x times x 3x^2). Since there are two ends, the total end area is (2 times 3x^2 6x^2).

Total internal surface area: The sum of these areas is (18x^2 18x^2 6x^2 36x^2).

Solving for Depth (x)

Given that the internal surface area is 158.76 m2, we can set up the equation:

[36x^2 158.76]

Solving for (x^2), we get:

[x^2 frac{158.76}{36} 4.41]

Therefore,

[x sqrt{4.41} 2.1]

The depth of the pool is approximately 2.1 meters.

Conclusion

By understanding the relationship between the width, length, and depth of a rectangular pool and the total internal surface area, we can derive the depth of the pool. This problem demonstrates the importance of algebraic problem solving in real-world applications, such as pool design and maintenance.