Understanding the Equation of a Sphere

The video tutorial explains how to find the standard equation of a sphere given the endpoints of its diameter. It covers calculating the center using the midpoint formula, determining the radius by finding the length of the diameter, and forming the sphere's equation. The tutorial concludes with a graphical representation of the sphere and its diameter.

What is the first step in finding the equation of a sphere given the endpoints of a diameter?

• Calculate the radius directly.

• Graph the endpoints.

• Find the midpoint of the diameter.

• Use the distance formula.

How do you determine the x-coordinate of the center of a sphere?

• Add the x-values of the endpoints and divide by 2.

• Divide the x-values of the endpoints.

• Multiply the x-values of the endpoints.

• Subtract the x-values of the endpoints.

What is the y-coordinate of the center if the endpoints are (4, 5, 2) and (3, 1, -6)?

• 5

• 2

• 4

• 3

Why is it beneficial to calculate the diameter first when finding the radius?

• It is a requirement.

• It is more accurate.

• It is faster.

• It avoids using decimals.

What is the length of the diameter if the endpoints are (4, 5, 2) and (3, 1, -6)?

• 7

• 5

• 9

• 11

What is the radius of the sphere if the diameter is 9?

• 4.5

• 3.5

• 5.5

• 6.5

Which of the following represents the equation of a sphere?

• (x - 3.5)^2 + (y - 3)^2 + (z + 2)^2 = 20.25

• (x + 3.5)^2 + (y + 3)^2 + (z - 2)^2 = 20.25

• (x - 3.5)^2 + (y + 3)^2 + (z - 2)^2 = 20.25

• (x + 3.5)^2 + (y - 3)^2 + (z + 2)^2 = 20.25

What is the value of R^2 in the sphere's equation if the radius is 4.5?

• 20.25

• 24.25

• 18.25

• 22.25

What does the graphical representation of the sphere help illustrate?

• The equation of the sphere.

• The center and radius of the sphere.

• All of the above.

• The endpoints of the diameter.

What is the significance of the black point in the graphical representation?

• It represents the radius.

• It represents the center of the sphere.

• It represents an endpoint.

• It represents the diameter.