Calculus III: Three Dimensional Coordinate Systems (Level 7 of 10)

The video tutorial explains how to derive the equation of a sphere using three examples. The first example involves finding the equation of a sphere given its radius and center. The second example demonstrates how to find the equation when the center and a point on the sphere are provided. The third example shows how to determine the equation using the endpoints of the diameter. Each example involves using the distance formula and substituting values into the sphere equation.

What is the equation of a sphere with a center at (2, -1, 3) and a radius of 5?

• (x - 2)^2 + (y - 1)^2 + (z - 3)^2 = 5

• (x + 2)^2 + (y + 1)^2 + (z + 3)^2 = 25

• (x + 2)^2 + (y - 1)^2 + (z + 3)^2 = 5

• (x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 25

When given the center and a point on the sphere, how do you find the radius?

• Use the Pythagorean theorem

• Use the area formula

• Use the distance formula

• Use the midpoint formula

What is the equation of a sphere with center (3, 8, 1) and passing through the point (4, 3, -1)?

• (x - 3)^2 + (y + 8)^2 + (z - 1)^2 = 30

• (x + 3)^2 + (y + 8)^2 + (z + 1)^2 = 30

• (x - 3)^2 + (y - 8)^2 + (z - 1)^2 = 30

• (x + 3)^2 + (y - 8)^2 + (z + 1)^2 = 30

How do you find the center of a sphere when given the endpoints of its diameter?

• Use the volume formula

• Use the distance formula

• Use the midpoint formula

• Use the area formula

What is the equation of a sphere with endpoints of the diameter at (7, -1, -2) and (3, 4, 6)?

• (x + 5)^2 + (y - 3/2)^2 + (z + 2)^2 = 105/4

• (x - 5)^2 + (y + 3/2)^2 + (z - 2)^2 = 105/4

• (x + 5)^2 + (y + 3/2)^2 + (z + 2)^2 = 105/4

• (x - 5)^2 + (y - 3/2)^2 + (z - 2)^2 = 105/4

What is the relationship between the diameter and radius of a sphere?

• Diameter is three times the radius

• Diameter is equal to the radius

• Diameter is twice the radius

• Diameter is half the radius

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

• Find the surface area

• Calculate the radius

• Find the midpoint of the diameter

• Calculate the volume