Question

Write the inequalities to describe the region of a solid cylinder that lies on or below the plane \(z = 8\) and on or above the disk in the \(xy\)-plane with the center at the origin and radius 2.

Short answer

The inequalities to describe the region of a solid cylinder that lies on or below the plane \(z = 8\) and on or above the disk in the \(xy\)-plane with the center at the origin and radius 2 is \({x^2} + {y^2} \le 4,0 \le z \le 8\).

Step-by-step solution

Simplify the equation.

\({R^3}\)is the three-dimensional coordinate system which contains\(x\),\(y\),and\(z\)-coordinates.

The equation \(z = 8\) in \({R^3}\) represents the set \(\{ (x,y,z)\mid z = 8\} \), which is the set of all points in \({R^3}\) whose \(z\) coordinate is 8 and \(x\), \(y\)-coordinates are any values.

The inequality to describe the region in which all points lie on or below the plane \(z = 8\) is \(0 \le z \le 8\).

The equation to describe the region of cylinder with center at the origin and radius of 2 units on the \(xy\)-plane is \({x^2} + {y^2} = 4\).But, it is required to describe the region in which all points lie on or above the disk in the \(xy\)- plane.

The inequality to describe the region of solid cylinder that lies on or above the disk in the \(xy\)-plane with center at origin and radius of 2 units is \({x^2} + {y^2} \le 4\).

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Thus, the inequalities to describe the region of a solid cylinder that lies on or below the plane \(z = 8\) and on or above the disk in the \(xy\)-plane with the center at the origin and radius 2 is \({x^2} + {y^2} \le 4,0 \le z \le 8\).

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