Question

An elliptical-shaped path surrounds a garden, modeled by quantity x minus 20 end quantity squared over 169 plus quantity y minus 18 end quantity squared over 289 equals 1 comma where all measurements are in feet. What is the maximum distanc

Answer 1

The maximum distance between any two points of the ellipse is 34 feet.

Procedure - Determination of the distance between two points of a ellipse

The maximum distance between any two points of a ellipse is the maximum distance between the ends of the ellipse along the longest axis, which is parallel to the y-axis in this case.

In addition, the equation of the ellipse in standard form is defined by this formula:

$\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1$ (1)

Where:

• $h, k$ - Coordinates of the center of the ellipse.
• $a$, $b$ - Lengths of each semiaxis.

Hence, the maximum distance ($d_{max}$), in feet, is calculated by this formula:

$d_{max} = 2\cdot b$ (2)

If we know that $b = 17$, then the maximum distance between any two points of the ellipse is:

$d_{max} = 2\cdot (17\,ft)$

$d_{max} = 34\,ft$

The maximum distance between any two points of the ellipse is 34 feet. $\blacksquare$

Remark

The statement is incomplete and poorly formatted, correct form is presented below:

An elliptical-shaped path surrounds a garden, modeled by $\frac{(x-20)^{2}}{169} + \frac{(y-18)^{2}}{289} = 1$, where all measurements are in feet. What is the maximum distance between any two points of the path.

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