Question

To determine the surface of the equation \(z = 4 - {r^2}\).

Short answer

The surface of \(z = 4 - {r^2}\) is a circular paraboloid.

Step-by-step solution

Given data

The equation \(z = 4 - {r^2}\).

Formula used resultant

The formula to find \(r\) from the rectangular coordinates \((x,y,z)\) is \(r = \sqrt {{x^2} + {y^2}} \).

Use formula and solve

The formula to find \(r\) from the rectangular coordinates \((x,y,z)\) is \(r = \sqrt {{x^2} + {y^2}} \).

From the formula mentioned above, it is obtained that \({x^2} + {y^2} = {r^2}\).

Therefore, the given equation becomes,

\(\begin{array}{c}z &=& 4 - \left( {{x^2} + {y^2}} \right){x^2} + {y^2}\\ &=& 4 - z\end{array}\)

which is the equation of a circular paraboloid.

Therefore, the given surface is a circular paraboloid.