Converting Cylindrical Equations to Rectangular Form

The video tutorial explains how to convert cylindrical equations to rectangular form using four examples. It covers the necessary substitutions for variables R and θ to express equations in terms of x, y, and z. Each example demonstrates a different cylindrical equation and its conversion process, highlighting the use of trigonometric identities and algebraic manipulation to achieve the rectangular form.

What are the forms of cylindrical and rectangular coordinates?

• Cylindrical: (x, y, z), Rectangular: (r, theta, z)

• Cylindrical: (r, theta, z), Rectangular: (x, y, z)

• Cylindrical: (theta, r, z), Rectangular: (y, x, z)

• Cylindrical: (r, z, theta), Rectangular: (x, z, y)

In the first example, what substitution is made to convert z = r^2 * cos^2(theta) to rectangular form?

• r = sqrt(x^2 + y^2)

• y = r * sin(theta)

• theta = arctan(y/x)

• x = r * cos(theta)

What is the rectangular form of the equation z = r^2 * cos^2(theta)?

• z = r^2

• z = x^2 + y^2

• z = x^2

• z = y^2

What shape does the equation r^2 + z^2 = 16 represent in rectangular coordinates?

• A cylinder

• A cone

• A sphere

• A paraboloid

What is the radius of the sphere represented by the equation r^2 + z^2 = 16?

• 2

• 4

• 5

• 3

In the third example, what substitution is used for sin(theta) in the equation r = 2 * sin(theta)?

• sin(theta) = z/r

• sin(theta) = x/r

• sin(theta) = y/r

• sin(theta) = r/y

What is the rectangular form of the equation r = 2 * sin(theta)?

• x^2 + y^2 = 4

• x^2 + y^2 = 2x

• x^2 + y^2 = 2z

• x^2 + y^2 = 2y

In the fourth example, what is the rectangular form of the equation r = 10z?

• x^2 + y^2 = 10

• x^2 + y^2 = 10z

• x^2 + y^2 = 100z^2

• x^2 + y^2 = z^2

What substitution is made for r in the equation r = 10z?

• r = x + y

• r = x^2 + y^2

• r = sqrt(x^2 + y^2)

• r = z^2

Which of the following equations is in rectangular form?

• x^2 + y^2 = 100z^2

• r^2 + z^2 = 16

• r = 10z

• z = r^2 * cos^2(theta)