Question

Find a parametric representation for the surface: the part of the plane z=x3 that lies inside the cylinder x2+y2 =1.

A) x=cos(t),y=sin(t),z=cos3(t)
B) x=sin(t),y=cos(t),z=sin3(t)
C) x=cos(t),y=sin(t),z=sin3(t)
D) x=sin(t),y=cos(t),z=cos3(t)

Answer 1

Option A) x=cos(t), y=sin(t), z=cos^3(t) provides the correct parametric representation for the surface z=x^3 that lies inside the cylinder x^2+y^2=1, as it satisfies the constraints of both the plane and the cylinder.Therefore, the correct parametric equations that satisfy both the surface and the cylinder are x=cos(t), y=sin(t), z=cos^3(t), which corresponds to option A).

Step-by-step explanation:

The student has asked to find a parametric representation for the surface of the plane z=x^3 that lies inside the cylinder x^2+y^2=1. The correct parametric equations for the surface must satisfy both the equation of the plane and the equation of the cylinder.

Let's examine the options given:

x=cos(t), y=sin(t) ensures that x^2 + y^2 = cos^2(t) + sin^2(t) = 1, which satisfies the cylinder constraint.

To find a parametric representation for the part of the plane z = x^3 that lies inside the cylinder x^2 + y^2 = 1, we need to express x, y, and z in terms of a single parameter, let’s call it t.

For the cylinder x^2 + y^2 = 1, we can parameterize the x and y coordinates using trigonometric functions that satisfy this equation. Using the unit circle as a guide, we know that an angle t measured in radians from the positive x-axis around the circle gives the coordinates (cos(t), sin(t)) on the circle.

Thus, we can parameterize the coordinates (x, y) of the cylinder as follows: - x = cos(t) - y = sin(t) Where t is the parameter that varies and determines the position on the circle (or cylinder, in the context of three dimensions).

Now, for the plane z = x^3, we want to apply this parameterization to the x-coordinate so we can express z in terms of t as well. Substituting the parameterized x into the equation for z gives us: - z = cos(t)^3 or, more simply, z = cos^3(t) Putting these together, we get the parametric equations for the surface that is part of the plane inside the cylinder: - x = cos(t) - y = sin(t) - z = cos^3(t) Therefore, the correct parametric representation for the surface in question is: - x = cos(t) - y = sin(t) - z = cos^3(t) This corresponds to option (A).

For the plane z=x^3, we need to replace x in the plane's equation with the parametric x(t). Thus, z should be cos^3(t).

Therefore, the correct parametric equations that satisfy both the surface and the cylinder are x=cos(t), y=sin(t), z=cos^3(t), which corresponds to option A).