Question

You have 1000 feet of flexible pool siding and intend to construct a swimming pool. Experiment with rectangular-shaped pools with perimeters of 1000 feet. How do their areas vary? What is the shape of the rectangle with the largest area? Now compute the area enclosed by a circular pool with a perimeter (circumference) of 1000 feet. What would be your choice of shape for the pool? If rectangular, what is your preference for dimensions? Justify your choice. If your only consideration is to have a pool that encloses the most area, what shape should you use?

Short answer

The circular pool, with approximately `79617.4` square feet, encloses the most area.

Step-by-step solution

Understanding the perimeter constraint

For a rectangular pool, the perimeter is given by the formula: `Perimeter = 2 * length + 2 * width`. Since the perimeter is 1000 feet, we have `1000 = 2 * length + 2 * width` or `500 = length + width`.

Express width in terms of length

From the equation `500 = length + width`, we can express the width as `width = 500 - length`.

Derive the area formula

The area of a rectangle is given by `Area = length * width`. Substituting `width = 500 - length` into this formula, we get: `Area = length * (500 - length)` or `Area = 500 * length - length^2`.

Maximizing the area

To find the maximum area, take the derivative of `Area = 500 * length - length^2` with respect to `length` and set it to zero: `d(Area)/d(length) = 500 - 2 * length = 0`, solving for `length` gives `length = 250`.

Determine the dimensions and area

Substitute `length = 250` back into `width = 500 - length` to get `width = 250`. Thus, the rectangle with the largest area is a square with side lengths of 250 feet. The area is `Area = 250 * 250 = 62500` square feet.

Circular pool area

For a circular pool with circumference 1000 feet, the circumference formula is `Circumference = 2 * π * radius = 1000`. Solving for the radius gives `radius = 1000 / (2 * π) = 500 / π`.The area of the circle is `Area = π * radius^2 = π * (500 / π)^2 = 250000 / π` square feet or approximately `79617.4` square feet.

Compare the areas and make a choice

Comparing the areas, the circular pool has approximately `79617.4` square feet while the maximum rectangular (square) pool has `62500` square feet. Thus, the circular pool encloses a larger area.

Key concepts

Rectangular Pool Area Maximization

Constructing a pool with a specified perimeter means we need to understand how shape affects the area we can enclose. Here, we're examining a rectangular pool with a 1000-foot perimeter. The formula for the perimeter of a rectangle is:
Perimeter = 2 * length + 2 * width
Given that the perimeter is 1000 feet, we can say:
500 = length + width (rearranged for simplicity).
To maximize the area, we need to express width in terms of length:
width = 500 - length
The area of a rectangle is given by:
Area = length * width
Substituting width, we get:
Area = length * (500 - length)
To find the maximum area, we derive the area equation with respect to length and set it to zero:
d(Area)/d(length) = 500 - 2 * length = 0
Solving this, we get length = 250 feet and subsequently, width = 250 feet. This shows that the maximum area within the constraint is a square with side lengths of 250 feet, having the formula:
Area = 250 * 250 = 62500 square feet.
Hence, through calculus and understanding geometric properties, the optimal rectangle for maximum area is indeed a square.

Circular Pool Area Calculation

Next, consider constructing a circular pool with the same perimeter of 1000 feet. The circumference (perimeter) of a circle is given by the formula:
Circumference = 2 * π * radius
Given that Circumference = 1000 feet, we can solve for the radius:
radius = 1000 / (2 * π) = 500 / π
The area of a circle is given by:
Area = π * radius^2
Substituting the calculated radius, we get:
Area = π * (500 / π)^2 = 250000 / π
which is approximately 79617.4 square feet.
Practically, this means that a circular pool with a 1000-foot circumference encloses a considerably larger area compared to a rectangular pool, confirming that circles are the most area-efficient shape given a fixed perimeter.

Derivatives in Optimization

The concept of using derivatives is crucial in optimization problems, such as maximizing the area of a pool with a given perimeter. Derivatives help us find the rate at which a function changes, allowing us to determine maxima and minima. The steps are as follows:

• Express the function in terms of one variable, such as length in our rectangular pool example.
• Derive the function with respect to this variable.
• Set the derived function (first derivative) to zero to find critical points.
• Solve for the variable to find the possible points that maximize or minimize the function.

For the rectangle, we derived the area function, found its first derivative, and set it to zero:
d(Area)/d(length) = 500 - 2 * length = 0
Solving gave us the optimal length, which we used to calculate the width and area, leading us to find that a square yields the maximum area. Similar methods are used in various optimization problems beyond geometry.

Geometric Properties

Understanding geometric properties allows us to better maneuver within given constraints, like perimeter in this exercise. Properties of shapes, such as rectangles and circles, show how area behaves with changed dimensions. The core geometric facts we used are:

• A square is a rectangle with equal side lengths, and it maximizes the area for a given perimeter.
• The area of a shape is influenced by the relationship between its dimensions.
• A circle, for any given perimeter, encloses the maximum area of all plane shapes due to the nature of π.

Among all shapes, a circle is the most area-efficient. For any constrained perimeter, a circle encloses the largest area. This principle is vital for applications in design, architecture, and various fields requiring optimal space utilization. Understanding these geometric properties not only helps in solving textbook problems but also in real-life practical scenarios.