Problem 59 - Intermediate Algebra

Question

A landscape architect is designing an elliptical fish pond that will fit in the center of a 110 -by- 100 -foot rectangular Japanese rock garden, leaving 15 feet of clearance on all sides. If she establishes a coordinate system with the 110 -foot length along the \(x\) -axis, and the center of the pond at the origin, find the equation of the ellipse.

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{1600} + \frac{y^2}{1225} = 1 \).

Step 1: Determine the Dimensions of the Pond

The pond leaves a 15-foot clearance on each side of the rectangle, so the width of the pond in the x direction is given by subtracting twice the clearance from the total length (110 feet). This gives a width: 110 - 2(15) = 80 feet. Similarly, the height along the y-axis is: 100 - 2(15) = 70 feet.

Step 2: Establish the Semi-Axes

Since the elliptical pond is centered at the origin, half of the width (80 feet) is the semi-major axis along the x-axis and half of the height (70 feet) is the semi-minor axis along the y-axis. Therefore, the semi-major axis is 40 feet, and the semi-minor axis is 35 feet.

Step 3: Write the Equation of the Ellipse

The standard form for the equation of an ellipse centered at the origin with the semi-major axis on the x-axis and semi-minor axis on the y-axis is: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a = 40\) and \(b = 35\). Thus, the equation becomes: \( \frac{x^2}{1600} + \frac{y^2}{1225} = 1 \).

Key Concepts

Understanding the Coordinate System for Ellipses

When working with ellipses in geometry, it's crucial to understand the coordinate system. A coordinate system allows you to establish positions using numerical coordinates, typically along two axes: the x-axis and the y-axis. In the context of the fish pond problem, a coordinate system has been established where the center of the ellipse coincides with the origin of the plane. This makes calculations more straightforward as it simplifies the equation of the ellipse.

By orientating the longer length (110 feet) of the rectangular garden along the x-axis, we align the major dimension of the elliptic pond with this axis. Similarly, the shorter width (100 feet) aligns with the y-axis. This setup helps in fitting the ellipse perfectly within the given garden while ensuring that the pond is centered, forming an even gap around all edges.

Defining the Semi-Major Axis

The term 'semi-major axis' refers to half of the longest diameter of an ellipse. In simpler terms, it's the largest radius that stretches from the center of the ellipse to the outer edge, cutting across its widest point. For the elliptical pond at the center of this Japanese rock garden, once the clearances are accounted for, the length remaining (110 - 2 times the 15-foot clearance) is 80 feet, resulting in a semi-major axis of 40 feet.

Remember:

• Semi-major axis length is a critical factor in determining the overall shape and size of the ellipse.
• It helps define part of the standard equation of the ellipse, conveyed as the variable \(a\).
• The semi-major axis always aligns with the x-axis if it’s the longer axis, which it is in this example.

Understanding the Semi-Minor Axis

The semi-minor axis of an ellipse is half of its shortest diameter. Just like the semi-major axis, this axis stretches from the center to the boundary but through the narrowest part instead. Within the context of the elliptical fish pond, the dimension along the y-axis after deducting the required clearances (100 - 2 times the 15-foot clearance) is 70 feet, giving a semi-minor axis of 35 feet.

Key points about the semi-minor axis:

• It is referred to as \(b\) in the standard ellipse equation.
• Smaller than the semi-major axis, it typically aligns with the y-axis.
• Its role is vital to achieving the elliptical shape, impacting how round or elongated the ellipse appears.

The Geometry of Ellipses

Geometry involves studying shapes, sizes, and the properties of space. Ellipses are fascinating as they belong to a specific type of geometry called conic sections, formed by slicing through a cone at an angle. Understanding geometry's role in this problem aids in visualizing and solving it effectively.

An ellipse is defined as the collection of all points where the sum of the distances from two fixed points, known as the foci, remains constant. In the case of the fish pond, the semi-major and semi-minor axes help frame this overall shape. The equation reflecting this form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively. Utilizing these parameters, one can calculate precise dimensions and construct the ellipse to fit within the designated garden space while respecting the given clearances.