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Chapter 15: Problem 7

Without calculus, find the highest and lowest points (if they exist) on the surface. The z-axis is upward. $$z=(x-5)^{2}+(y-\pi)^{2}+2 \pi$$

Short Answer

Expert verified
The minimum \(z\) is \(2\pi\) at \((x, y) = (5, \pi)\); no maximum exists.

Step by step solution

01

Understand the Structure

The given surface equation is \( z = (x-5)^2 + (y-\pi)^2 + 2\pi \). This is the equation of a paraboloid. It can be seen as a 2D parabola extended in all directions, forming a bowl shape.
02

Identify the Center of the Paraboloid

Examining the equation \((x-5)^2 + (y-\pi)^2\), it's clear that the vertex (or center) of the paraboloid in the \(xy\)-plane is at \((x, y) = (5, \pi)\). This is the point where \((x-5)\) and \((y-\pi)\) both equal zero, minimizing \(z\).
03

Determine Minimum z Value

At the point \((x, y) = (5, \pi)\), the equation simplifies to \(z = (5-5)^2 + (\pi-\pi)^2 + 2\pi = 2\pi\). Therefore, the minimum value of \(z\) on this surface is \(2\pi\).
04

Explore Maximum Possibilities

Since the equation \(z = (x-5)^2 + (y-\pi)^2 + 2\pi\) defines an upward-opening paraboloid, \(z\) continues to increase as \(x\) or \(y\) move away from the center. There is no highest point; \(z\) can be arbitrarily large.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Paraboloid
In the study of paraboloids, the vertex is a crucial point to identify. This is where the paraboloid is at its most balanced state in the 3D space. For the equation provided in the exercise, \( z = (x-5)^2 + (y-\pi)^2 + 2\pi \), the vertex is located by examining the expressions
  • \((x-5)^2\)
  • \((y-\pi)^2\)
To determine the vertex coordinates
  • Set each squared term to zero
  • This makes both terms equal to zero at \(x = 5\) and \(y = \pi\)
Thus, the vertex of the paraboloid is at the point \((5, \pi)\) in the \(xy\)-plane. At this point, the equation simplifies and shows where the surface of the paraboloid reaches its minimum height (along the z-axis). Understanding the vertex helps in understanding the geometry and symmetries of the paraboloid.
Minimum Value of a Function
The minimum value of a function often indicates the lowest point of a surface in relation to the chosen coordinate system. For our paraboloid, the minimum value of \(z\) occurs at the vertex, \((5, \pi)\). By substituting these values into the equation,
  • \(z = (5-5)^2 + (\pi-\pi)^2 + 2\pi\)
  • This calculates as \(z = 0 + 0 + 2\pi\)
  • Thus, \(z = 2\pi\)
This means the surface of the paraboloid, represented by the function, reaches its minimum height of \(2\pi\) units above the origin in the \(z\)-direction. Finding the minimum value is key to understand how the function behaves at its lowest point, and knowing where it sits in the space helps predict how it might extend or expand as the variables change.
Upward-Opening Paraboloid
An upward-opening paraboloid is a three-dimensional shape extending infinitely upwards, forming a bowl-like structure. Recognizing its behavior in the provided equation
  • The paraboloid is described by \( z = (x-5)^2 + (y-\pi)^2 + 2\pi \)
  • Both \((x-5)^2\) and \((y-\pi)^2\) contribute positively to \(z\)
This indicates as \(x\) or \(y\) increase or decrease away from their center values (5 and \(\pi\) respectively),
  • Changes in \(x\) or \(y\) result in larger \((x-5)^2\) and \((y-\pi)^2\)
  • This leads to higher \(z\) values
Consequently, there is no maximum value for \(z\); the surface grows to infinity upwards as you move further from the vertex. Recognizing the direction of opening helps in anticipating how the paraboloid spans out in three-dimensional space, critical for understanding its infinite or finite characteristics.

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