91Ó°ÊÓ

Cylindrical coordinates are a 3-dimensional coordinate system consisting of the variables (r, θ, z), where r is the radial distance from the z-axis, θ is the angle with respect to the positive x-axis in the xy-plane, and z is the height along the z-axis. Now let's analyze and sketch the object with the given constraints:

The radial distance r has a constraint, which states that \(0 \leq r \leq 3\). This means that from the z-axis, the distance of any point should not exceed 3 units in any radial direction. This basically creates a cylinder of radius 3 units with infinite height, since there is no constraint on the height yet.

The angular constraint is given by \(0 \leq θ \leq \frac{\pi}{3}\). This restricts the object to have points only within the angular range of \(0\) to \(\frac{\pi}{3}\) radians (or \(0\) to \(60^{\circ}\)) counterclockwise with respect to the positive x-axis in the xy-plane. This angular limitation will effectively carve out a pie-slice shape with a \(60^{\circ}\) opening from the cylindrical object that we have from the previous constraint. But the height is still unlimited.

The height constraint states that \(1 \leq z \leq 4\). This further restricts our object, as it implies that the object has a minimum height of 1 unit and a maximum height of 4 units from the xy-plane. It essentially cuts the top and the bottom of our pie-slice shape at z = 1 and z = 4, leaving a finite height for our object.

With all three constraints considered, our final object will resemble a pie-shaped cylindrical wedge with a 60-degree opening angle, a radius of 3 units, and a height restricted between z = 1 and z = 4.

To sketch this object, follow these steps: 1. Draw the xy-plane with axes x and y. Mark the positive x-axis as the angle θ = 0 direction. 2. Draw a vertical line (z-axis) passing through the point where x and y axes intersect. 3. Draw a circle of radius 3 units in the xy-plane centered at the z-axis. 4. Package the circle by two lines separated by an angle of 60 degrees, with one line along the positive x-axis and the other line at a 60-degree angle counterclockwise to the x-axis. 5. Extrude this 60-degree-sector of circle up to a height of 4 units from the xy-plane. 6. Draw a horizontal plane at z = 1 which cuts the pie-shaped cylindrical wedge. 7. The final sketch should resemble a 3D pie-shaped cylindrical wedge with a 60-degree opening angle, a radius of 3 units, and a height restricted between z = 1 and z = 4.