Questions: What is (are) the value(s) of x at the vertex (vertices) of the shape defined by the equation x^2/64 + y^2/36 = 1? (Separate multiple values of x with a comma.)

What is (are) the value(s) of x at the vertex (vertices) of the shape defined by the equation x^2/64 + y^2/36 = 1? (Separate multiple values of x with a comma.)
Transcript text: What is (are) the value(s) of $x$ at the vertex (vertices) of the shape defined by the equation $\frac{x^{2}}{64}+\frac{y^{2}}{36}=1$ ? (Separate multiple values of $x$ with a comma.)
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Solution

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Solution Steps

Step 1: Identify the Equation of the Ellipse

The given equation is x264+y236=1\frac{x^2}{64} + \frac{y^2}{36} = 1. This is the standard form of an ellipse centered at the origin (0,0)(0, 0) with semi-major and semi-minor axes.

Step 2: Determine the Semi-Major and Semi-Minor Axes

In the standard form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, the values a2=64a^2 = 64 and b2=36b^2 = 36 are given. The semi-major axis is along the x-axis because a2>b2a^2 > b^2.

Step 3: Calculate the Length of the Semi-Major Axis

The length of the semi-major axis is given by a=64=8a = \sqrt{64} = 8.

Step 4: Identify the x-Coordinates of the Vertices

The vertices of the ellipse along the x-axis are located at (±a,0)(\pm a, 0). Therefore, the x-coordinates of the vertices are x=8x = -8 and x=8x = 8.

Final Answer

The x-coordinates of the vertices are 8,8\boxed{-8, 8}.

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