Problem 8 - Calculus Concepts

Question

Geometry The equation \(y=\pm \sqrt{36-x^{2}}\) is the \(y\) -coordinate for a point with \(x\) -coordinate between -6 and 6 on a circle with radius \(6 \mathrm{~cm}\) centered at the origin.

Step-by-Step Solution

Answer

The equation describes the y-coordinates of a circle with radius 6 cm centered at the origin.

Step 1: Understanding the Equation of a Circle

The general equation for a circle centered at the origin is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. In this problem, the radius \(r\) is given as 6 cm.

Step 2: Substituting 6 for Radius

Given that the radius \(r\) is 6, substitute this value into the equation \(x^2 + y^2 = r^2\), resulting in \(x^2 + y^2 = 36\).

Step 3: Solving for y

To solve for \(y\), subtract \(x^2\) from both sides of the equation \(x^2 + y^2 = 36\) to get \(y^2 = 36 - x^2\).

Step 4: Taking Square Roots

Take the square root of both sides of the equation \(y^2 = 36 - x^2\) to solve for \(y\). This gives \(y = \pm \sqrt{36 - x^2}\).

Step 5: Interpreting the +/- Symbol

The \(\pm\) symbol indicates that for a given \(x\)-coordinate, there are two \(y\)-coordinates: one positive and one negative, which correspond to the points above and below the x-axis on the circle.

Step 6: Considering the Range of x-values

The problem states that the \(x\)-coordinate is between -6 and 6. This is consistent with the definition of the circle's boundary, confirming the equation's validity.

Key Concepts

Radius of a Circle

The radius of a circle is a crucial component in understanding its properties and equation. A circle is defined as all the points in a plane that are at a constant distance, known as the radius, from a fixed point, the center. In a circle centered at the origin, the radius determines the size and scale of the circle.

For example, if we have the equation of a circle given as \(x^2 + y^2 = r^2\), the term \(r^2\) represents the square of the radius. From this, we can deduce that the radius \(r\) is the square root of this term. In our specific problem, with the equation \(x^2 + y^2 = 36\), we can directly calculate the radius by noting that \(36\) is the square of our radius; hence, the radius \(r\) is 6 cm.

Understanding the radius helps us visualize the extent of the circle on the coordinate plane and is pivotal when graphing the circle or considering any geometric properties related to it.

Center of a Circle

The center of a circle is another key element that defines its position in a plane. When describing a circle, the center provides a fixed point from which all points on the circle maintain a consistent distance, which is the radius.

In a standard equation form for circles, \(x^2 + y^2 = r^2\), the circle is centered at the origin, which is the point (0, 0). This makes calculations simpler and visuals clearer because all transformations revolve around this central point.

Knowing the center of a circle allows us to understand its symmetry, as every point on the circle is equidistant to this central point. For circles not centered at the origin, an expanded form of the equation \((x - h)^2 + (y - k)^2 = r^2\) is used, where \((h, k)\) corresponds to the circle's center. However, in our current context, the simplicity of a center at the origin simplifies computations and provides a clear baseline for understanding circle equations.

Range of x-values

Determining the range of x-values is essential when working with circle equations, especially if we are interested in specific segments or sections of the circle.

The equation \(y = \pm \sqrt{36 - x^2}\) defines the y-coordinates derived from the circle equation \(x^2 + y^2 = 36\). Importantly, the values \(x\) can assume are logically constrained by the term under the square root, \(36 - x^2\), which must be non-negative to yield real results.

Thus, we require \(36 - x^2 \geq 0\), leading to \(-6 \leq x \leq 6\). This inequality describes the full horizontal span of the circle, from one side to the other. Any x-value outside this range would lead to complex numbers, which have no graphical representation in standard Cartesian coordinates for our circle.

Understanding the range of x-values ensures that we accurately interpret and work with the segments of the circle relevant to our problem, providing confidence in our solution and graphical interpretations.