Calculus III: Three Dimensional Coordinate Systems (Level 9 of 10)

The video tutorial covers two main problems involving spheres in three-dimensional coordinate systems. The first problem involves finding the largest sphere with a given center that is contained in the first octant. The solution involves expanding the sphere until it touches the closest coordinate plane. The second problem involves finding a sphere with its center in the XZ plane and passing through three given points. The solution requires setting up and solving a system of equations to find the center and radius of the sphere.

What is the primary goal when finding the largest sphere with a given center in the first octant?

• To minimize the sphere's radius.

• To expand the sphere until it touches the nearest coordinate plane.

• To place the sphere symmetrically in the octant.

• To ensure the sphere is tangent to all three coordinate planes.

How is the radius of the largest sphere in the first octant determined?

• By using the midpoint formula.

• By measuring the distance to the farthest coordinate plane.

• By finding the shortest distance from the center to a coordinate plane.

• By calculating the average distance to all coordinate planes.

In the problem of finding a sphere with its center in the XZ plane, what is known about the center's coordinates?

• The Y coordinate is zero.

• The Z coordinate is zero.

• The X coordinate is zero.

• The X and Y coordinates are known.

What method is used to find the radius of the sphere passing through points P, Q, and R?

• Using the midpoint formula.

• Equating the distances from the center to each point.

• Using the Pythagorean theorem.

• Calculating the area of the triangle formed by the points.

What is the first step in solving the system of equations to find the sphere's center?

• Equate the first and second expressions.

• Substitute known values into the equations.

• Eliminate the h squared term.

• Expand the binomial squared.

After finding the values of h and l, how is the radius of the sphere determined?

• By finding the midpoint of the line segments.

• By calculating the average of h and l.

• By using the distance formula with any of the three points.

• By using the Pythagorean theorem.

What is the final equation of the sphere located in the XZ plane?

• x^2 + y^2 + z^2 = 257

• (x - 5)^2 + (y - 4)^2 + (z - 9)^2 = 16

• (x + 7)^2 + y^2 + (z - 12)^2 = 257

• (x - 7)^2 + y^2 + (z + 12)^2 = 257