Find an equation of a sphere if one of its diameters has endpoints $(0, 3, 5)$ and $(4, 5, 7)$.

Neetesh Kumar | December 19, 2024

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This is the solution to Math 1C
Assignment: 12.1 Question Number 23
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Step-by-step solution:

Step 1: Finding the center of the sphere:

The center of the sphere is the midpoint of the diameter.

The midpoint formula for two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is:

$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$

Substitute the coordinates of the endpoints $(0, 3, 5)$ and $(4, 5, 7)$:

$\text{Center} = \left(\frac{0 + 4}{2}, \frac{3 + 5}{2}, \frac{5 + 7}{2}\right) = \left(2, 4, 6\right)$

Thus, the center of the sphere is $(2, 4, 6)$.

Step 2: Finding the radius of the sphere:

The radius is half the length of the diameter.

The formula for the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Substitute the coordinates of $(0, 3, 5)$ and $(4, 5, 7)$:

$\text{Diameter} = \sqrt{(4 - 0)^2 + (5 - 3)^2 + (7 - 5)^2}$

$\text{Diameter} = \sqrt{4^2 + 2^2 + 2^2} = \sqrt{16 + 4 + 4} = \sqrt{24}$

$\text{Radius} = \frac{\sqrt{24}}{2} = \sqrt{6}$

Step 3: Equation of the sphere:

The general equation of a sphere is:

$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$

where $(h, k, l)$ is the center and $r$ is the radius.

Substitute the center $(2, 4, 6)$ and radius $\sqrt{6}$:

$(x - 2)^2 + (y - 4)^2 + (z - 6)^2 = (\sqrt{6})^2$

Simplify:

$(x - 2)^2 + (y - 4)^2 + (z - 6)^2 = 6$

Final Answer:

The equation of the sphere is:

$\boxed{(x - 2)^2 + (y - 4)^2 + (z - 6)^2 = 6}$

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