Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Find the standard equation of the sphere. Endpoints of a diameter: \((1,0,0),(0,5,0)\)

Step-by-Step Solution

Answer

\((x - 0.5)^2 + (y - 2.5)^2 + z^2 = 26 / 4\)

Step 1: Finding the Center

The center of the sphere is the midpoint of the endpoints of the diameter. The midpoint between two points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) can be found using the formula \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})\). For the endpoints \((1,0,0)\) and \((0,5,0)\) of our diameter, this yields a center at \((\frac{1 + 0}{2}, \frac{0 + 5}{2}, \frac{0 + 0}{2}) = (0.5, 2.5, 0)\)

Step 2: Finding the Radius

The radius of the sphere is half the distance between the endpoints of the diameter. The distance between the points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). For our diameter's endpoints, this gives a radius of \(\sqrt{(0 - 1)^2 + (5 - 0)^2 + (0 - 0)^2} = \sqrt{26}\). Half of this is \(\sqrt{26} / 2\).

Step 3: Finding the Equation

Plugging the found center and radius values into the general equation of a sphere \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\), the equation of the sphere is \((x - 0.5)^2 + (y - 2.5)^2 + z^2 = (\sqrt{26} / 2)^2\). Simplifying the radius on the right hand side, we get the sphere's equation to be \((x - 0.5)^2 + (y - 2.5)^2 + z^2 = 26 / 4\).

Key Concepts

Understanding Geometry and Spheres

Geometry helps us explore the shapes, sizes, and positions of objects. A sphere is a perfect 3D object that is evenly round, like a ball or a globe. It has certain properties:

• The center is a point inside the sphere that is equidistant from all points on the surface.


• The radius is a line from the center to any point on the surface.


• The diameter is a line passing through the center and connecting two points on the surface; it is twice the radius.

In sphere problems, knowing the center and diameter is key to understanding its equation. This equation is a mathematical representation of all the points that form the sphere.

Using the Midpoint Formula

Calculating the midpoint is essential to find the center of the sphere when given a diameter. The midpoint formula is:\[\left( \frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}, \frac{{z_1 + z_2}}{2} \right)\]By averaging the x, y, and z coordinates of the endpoints, we determine the exact center. For the endpoints \((1,0,0)\) and \((0,5,0)\), the calculations are:

• x-component: \( \frac{1 + 0}{2} = 0.5 \)


• y-component: \( \frac{0 + 5}{2} = 2.5 \)


• z-component: \( \frac{0 + 0}{2} = 0 \)

So, the midpoint (center) is \((0.5, 2.5, 0)\). This point becomes the anchor for writing the sphere's equation.

Applying the Distance Formula

To find the radius of the sphere, we calculate the distance between the endpoints of the diameter using the distance formula:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]For the given endpoints, \((1,0,0)\) and \((0,5,0)\), the distance becomes:

• \((0 - 1)^2 = 1\)


• \((5 - 0)^2 = 25\)


• \((0 - 0)^2 = 0\)

Combining these, we get \(\sqrt{1 + 25 + 0} = \sqrt{26}\). The radius is half of this distance, which means \(\frac{\sqrt{26}}{2}\). This value is necessary to complete the sphere's equation.

Calculating the Radius and Writing the Sphere Equation

Now, with the center \((0.5, 2.5, 0)\) and the radius \(\frac{\sqrt{26}}{2}\), we use the standard sphere equation:\[(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\]Substitute the values:

• Center coordinates \(a = 0.5, b = 2.5, c = 0\)


• Radius \(r = \frac{\sqrt{26}}{2}\)

The equation becomes:\[(x - 0.5)^2 + (y - 2.5)^2 + z^2 = \left(\frac{\sqrt{26}}{2}\right)^2\]Simplify \[\left(\frac{\sqrt{26}}{2}\right)^2 = \frac{26}{4}\]So, the equation of the sphere is:\[(x - 0.5)^2 + (y - 2.5)^2 + z^2 = \frac{26}{4}\]This is a complete description of the sphere in mathematical terms.