/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 An einem Quader werden die L盲ng... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 1

An einem Quader werden die L盲ngen seiner Kanten und die Umf盲nge senkrecht zur ersten und zweiten Kante gemessen. Die Messwerte sind: Kante 1: \(26 \mathrm{~mm}\); Kante 2: \(38 \mathrm{~mm}\); Kante 3: \(55 \mathrm{~mm}\); Umfang \(\perp\) Kante 1: \(188 \mathrm{~mm}\); Umfang \(\perp\) K ante 2: \(163 \mathrm{~mm}\). Wie grofs sind die ausgeglichenen 碍补苍迟别苍濒盲苍驳别苍 nach der Methode der kleinsten Quadrate?

Short Answer

Expert verified
Adjusted lengths are \( a' = 34.5 \text{ mm}, b' = 47 \text{ mm}, c' = 47 \text{ mm} \).

Step by step solution

01

Understanding the Problem

We are given a cuboid with edge lengths (Kante) 1, 2, and 3 as 26 mm, 38 mm, and 55 mm, respectively. We also have measurements of the perimeters of the faces perpendicular to each of these edges. Our task is to find the adjusted edge lengths using the method of least squares.
02

Setting up the Equations for Perimeter

The face perpendicular to Kante 1 has the perimeter: \( 2b + 2c = 188 \). The face perpendicular to Kante 2 has the perimeter: \( 2a + 2c = 163 \). The third perimeter equation is implicitly given by the dimensions: \( 2a + 2b \).
03

Expressing the Problem as a System of Equations

We simplify the perimeter equations to express as: \( b + c = 94 \) and \( a + c = 81.5 \). For the method of least squares, considering measured values affording some error, we will treat these equations as identities we want to minimally adjust.
04

Applying Least Squares Adjustment

The least squares method requires minimizing the sum of squared differences between measured values and those predicted by our model. Assume true values \(a'\), \(b'\), \(c'\) as corrections to our measured data: 1. \( b' = b + e_b \), \( c' = c + e_c \) such that \( b' + c' = 94 \). Adjust to \( b' = \frac{94}{2} = 47 \), \( c' = \frac{94}{2} = 47 \). 2. \( a' + c' = 81.5 \). Adjust to \( c' = 47 \), thus \(a' = 81.5 - 47 = 34.5\).
05

Calculating Corrected Edge Lengths

The least squares adjustments give us the corrected edge lengths: - \( a' = 34.5 \text{ mm}, \ b' = 47 \text{ mm}, \ c' = 47 \text{ mm} \)
06

Verifying Solutions

Check if the new values meet the given perimeters: 1. \( 2b' + 2c' = 2(47) + 2(47) = 188\), which is satisfied for the perpendicular to Kante 1. 2. \( 2a' + 2c' = 2(34.5) + 2(47) = 163\), which is satisfied for the perpendicular to Kante 2. Thus, our least squares solution satisfies the conditions.

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Key Concepts

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

Kleinste-Quadrate-Methode
The "Kleinste-Quadrate-Methode," or the least squares method, is a mathematical approach used to find the best-fitting solution when data is subject to uncertainty or errors. It aims to minimize the sum of the squared differences between the observed and calculated values.

When dealing with a problem involving measurements, such as with a cuboid, measurements often have some error. This method helps in optimizing the calculated values to fit these measurements as closely as possible, effectively balancing the difference between observed and theoretical calculations.

The power of the least squares lies in its ability to convert potentially inaccurate measurements into a more reliable set of data.
Quader
A "Quader" is the German term for a cuboid or rectangular prism. It is a three-dimensional geometric figure with six rectangular faces, twelve edges, and eight vertices. Each face of the cuboid is a rectangle, and opposite faces are identical in shape and size.

In the problem, we are given the cuboid's edge lengths, which are crucial for calculating perimeters and carrying out adjustments using mathematical techniques such as the method of least squares. Understanding the geometric properties of a quader is necessary to correctly set up and solve equations related to its measurements.
碍补苍迟别苍濒盲苍驳别苍
"碍补苍迟别苍濒盲苍驳别苍" refers to the edge lengths of a geometric shape. Specifically, for a cuboid, these are the measurements of its three distinct edges. Each edge length is an important parameter that influences its volume and surface area.

In the exercise, the edge lengths are corrected using the least squares method due to measurement errors. The corrected edge lengths provide a more accurate depiction of the cuboid's true dimensions. By understanding and adjusting the kantenl盲ngen, one can ensure the cuboid maintains its geometric integrity while aligning closer to the given constraints.
Perimeter
The perimeter of a shape refers to the total length around a two-dimensional shape, and in this exercise, it pertains to the perimeters of the faces of the cuboid. Calculating the perimeter involves summing up the lengths of all sides of the particular face being considered.

In a three-dimensional context, like a cuboid, the perimeter measurement can help determine the relationships between different edge lengths, enabling formulas to be established. For example, the face perpendicular to edge 1 with a perimeter of 188 mm involves the other two edge lengths. Understanding these relationships is crucial for adjusting measurements effectively and ensuring accurate representation of geometric figures.
  • The formula for the perimeter of a rectangle is the sum of twice the width and twice the height.
  • It serves as important data for applying adjustments using mathematical tools like the least squares method.

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Most popular questions from this chapter

Um den Standort eines illegalen Senders festzustellen, werden f眉nf Peilwagen eingesetzt, mit denen die Richtungen zum Sender ermittelt werden. Die Aufstellung der Peilwagen ist in einem \((x, y)\)-Koordinatensystem gegeben, und die Richtungswinkel \(\alpha\) sind von der positiven \(x\) - Achse im Gegenuhrzeigersinn angegeben. $$ \begin{array}{|l|crccc|} \hline \text { Peilwagen } & 1 & 2 & 3 & 4 & 5 \\ \hline x \text {-Koordinate } & 4 & 18 & 26 & 13 & 0 \\ y \text {-Koordinate } & 1 & 0 & 15 & 16 & 14 \\ \text { Richtungswinkel } \alpha & 45^{\circ} & 120^{\circ} & 210^{\circ} & 270^{\circ} & 330^{\circ} \\ \hline \end{array} $$ Die Situation ist an einer grofen Zeichnung darzustellen. Welches sind die mutma脽lichen Koordinaten des Senders nach der Methode der kleinsten Quadrate? Als Startwert f眉r das Verfahren von Gau脽-Newton und f眉r Minimierungsmethoden w盲hle man beispielsweise \(P_{0}(12.6,8.0)\).

An einem Quader misst man die Kanten der Grundfl盲che \(a=21 \mathrm{~cm}, b=28 \mathrm{~cm}\) und die H枚he \(c=12 \mathrm{~cm} .\) Weiter erh盲lt man als Messwerte f茂r die Diagonale der Grundfl盲che \(d=34 \mathrm{~cm}\), f眉r die Diagonale der Seitenfl盲che \(e=24 \mathrm{~cm}\) und f眉r die K枚rperdiagonale \(f=38 \mathrm{~cm} .\) Zur Bestimmung der L盲ngen der Kanten des Quaders nach der Methode der kleinsten Quadrate verwende man das Verfahren von Gau脽-Newton und Minimierungsmethoden.

Die Konzentration \(z(t)\) eines Stoffes in einem chemischen Prozess gehorcht dem Gesetz $$ z(t)=a_{1}+a_{2} e^{\alpha_{1} t}+a_{3} e^{\alpha_{2} t}, \quad a_{1}, a_{2}, a_{3}, \alpha_{1}, \alpha_{2} \in \mathbb{R}, \quad \alpha_{1}, \alpha_{2}<0 $$ Zur Bestimmung der Parameter \(a_{1}, a_{2}, a_{3}, \alpha_{1}, \alpha_{2}\) liegen f眉r \(z(t)\) folgende Messwerte \(z_{k}\) vor. $$ \begin{array}{|l|lllllllll|} \hline t_{k}= & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 3.0 & 5.0 & 8.0 & 10.0 \\ \hline z_{k}= & 3.85 & 2.95 & 2.63 & 2.33 & 2.24 & 2.05 & 1.82 & 1.80 & 1.75 \\\ \hline \end{array} $$ Als Startwerte verwende man beispielsweise \(a_{1}^{(0)}=1.75, a_{2}^{(0)}=1.20, a_{3}^{(0)}=0.8, \alpha_{1}^{(0)}=-0.5\), \(\alpha_{2}^{(0)}=-2\) und behandle die nichtlinearen Fehlergleichungen mit dem Verfahren von Gau脽-Newton und mit Minimierungsmetho den.

Die Funktion \(y=\sin (x)\) ist im Intervall \([0, \pi / 4]\) durch ein Polynom \(P(x)=a_{1} x+\) \(a_{3} x^{3}\) zu approximieren, das wie \(\sin (x)\) ungerade ist. Die Koeffizienten \(a_{1}\) und \(a_{3}\) sind n ach der Methode der kleinsten Quadrate f眉r die diskreten St眉tzstellen \(x_{k}=k \pi / 24, k=1,2, \ldots, 6\), zu bestimmen. Mit dem gefundenen Polynom \(P(x)\) zeichne man den Graphen der Fehlerfunktion \(r(x):=P(x)-\sin (x)\)
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