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When dealing with geometric shapes like paraboloids and planes in 3D space, understanding how they intersect is crucial. An intersection in this context means finding where the plane cuts through the paraboloid. In our exercise, the plane is given by the equation \( z = 5 \), while the paraboloid is defined by \( z = 9 - x^2 - y^2 \). To find the line or curve at which these two shapes meet (i.e., intersect), we equate their equations: \[ 9 - x^2 - y^2 = 5 \]By setting the two equations equal, we are essentially "slicing" the paraboloid with the plane and observing the resultant shape on the \( xy \)-plane. Solving the equation results in: \[ x^2 + y^2 = 4 \]This equation is crucial as it defines the intersection as a circle with a specific property on the \( xy \)-plane.

The equation \( x^2 + y^2 = 4 \) represents a circle on the \( xy \)-plane. Here’s what you need to know about this circle:- **Center**: The center of the circle is at the origin, point (0,0).- **Radius**: The radius of this circle is 2, as indicated by the value on the right-hand side of the equation.This circle comes from the intersection of the plane and paraboloid. It showcases the boundary beyond which the paraboloid no longer exists above the plane.The circle equation is a fundamental result in understanding how 3D objects interact with each other. In simpler terms, drawing this circle out helps us visualize exactly where the paraboloid meets the plane.

Once we identify the intersection circle, the next step is to describe the 3D region that lies above the plane. In this context, the 3D region forms a segment of the paraboloid, called a cap, constrained between the paraboloid surface and the plane.- **Base**: The base of this cap is the circle we've just discussed, defined by \( x^2 + y^2 < 4 \). This means only points inside this circle on the \( xy \)-plane are part of our region of interest.- **Height**: The height begins at the plane \( z = 5 \) and extends upwards to the paraboloid \( z = 9 - x^2 - y^2 \).This cap represents all points that satisfy the paraboloid's constraint of having a \( z \)-value greater than 5. Visualizing this cap helps understand more complex geometrical relations and provides a clearer view of volume and space distributions in 3D shapes.