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Fynjy0 [20]
3 years ago
14

How do you figure out what numbers are terminating and what numbers are repeating ?

Mathematics
1 answer:
raketka [301]3 years ago
3 0
You can simply look and see if one is terminating by seeing if it ends. For example, 2.54 is terminating since the numbers after the decimal aren't repeating. A repeating number is for example 0.555555..
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Strain-displacement relationship) Consider a unit cube of a solid occupying the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 After loa
Anastasy [175]

Answer:

please see answers are as in the explanation.

Step-by-step explanation:

As from the data of complete question,

0\leq x\leq 1\\0\leq y\leq 1\\0\leq z\leq 1\\u= \alpha x\\v=\beta y\\w=0

The question also has 3 parts given as

<em>Part a: Sketch the deformed shape for α=0.03, β=-0.01 .</em>

Solution

As w is 0 so the deflection is only in the x and y plane and thus can be sketched in xy plane.

the new points are calculated as follows

Point A(x=0,y=0)

Point A'(x+<em>α</em><em>x,y+</em><em>β</em><em>y) </em>

Point A'(0+<em>(0.03)</em><em>(0),0+</em><em>(-0.01)</em><em>(0))</em>

Point A'(0<em>,0)</em>

Point B(x=1,y=0)

Point B'(x+<em>α</em><em>x,y+</em><em>β</em><em>y) </em>

Point B'(1+<em>(0.03)</em><em>(1),0+</em><em>(-0.01)</em><em>(0))</em>

Point <em>B</em>'(1.03<em>,0)</em>

Point C(x=1,y=1)

Point C'(x+<em>α</em><em>x,y+</em><em>β</em><em>y) </em>

Point C'(1+<em>(0.03)</em><em>(1),1+</em><em>(-0.01)</em><em>(1))</em>

Point <em>C</em>'(1.03<em>,0.99)</em>

Point D(x=0,y=1)

Point D'(x+<em>α</em><em>x,y+</em><em>β</em><em>y) </em>

Point D'(0+<em>(0.03)</em><em>(0),1+</em><em>(-0.01)</em><em>(1))</em>

Point <em>D</em>'(0<em>,0.99)</em>

So the new points are A'(0,0), B'(1.03,0), C'(1.03,0.99) and D'(0,0.99)

The plot is attached with the solution.

<em>Part b: Calculate the six strain components.</em>

Solution

Normal Strain Components

                             \epsilon_{xx}=\frac{\partial u}{\partial x}=\frac{\partial (\alpha x)}{\partial x}=\alpha =0.03\\\epsilon_{yy}=\frac{\partial v}{\partial y}=\frac{\partial ( \beta y)}{\partial y}=\beta =-0.01\\\epsilon_{zz}=\frac{\partial w}{\partial z}=\frac{\partial (0)}{\partial z}=0\\

Shear Strain Components

                             \gamma_{xy}=\gamma_{yx}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=0\\\gamma_{xz}=\gamma_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=0\\\gamma_{yz}=\gamma_{zy}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}=0

Part c: <em>Find the volume change</em>

<em></em>\Delta V=(1.03 \times 0.99 \times 1)-(1 \times 1 \times 1)\\\Delta V=(1.0197)-(1)\\\Delta V=0.0197\\<em></em>

<em>Also the change in volume is 0.0197</em>

For the unit cube, the change in terms of strains is given as

             \Delta V={V_0}[(1+\epsilon_{xx})]\times[(1+\epsilon_{yy})]\times [(1+\epsilon_{zz})]-[1 \times 1 \times 1]\\\Delta V={V_0}[1+\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}+\epsilon_{xx}\epsilon_{zz}+\epsilon_{yy}\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}\epsilon_{zz}-1]\\\Delta V={V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\

As the strain values are small second and higher order values are ignored so

                                      \Delta V\approx {V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\ \Delta V\approx [\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\

As the initial volume of cube is unitary so this result can be proved.

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2 years ago
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I’ll tell y’all 4 if did
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Anna's car requires 5 1/2 gallons of gasoline to make 4 round trips to work and back. If her car can travel 24 miles per gallon
marysya [2.9K]

Answer:

33 miles

Step-by-step explanation:

5.5÷4= 1.375 gal per round

1.375×24= 33 miles

4 0
2 years ago
Find the derivative of ln(cosx²).​
RoseWind [281]

Answer:

\displaystyle \frac{dy}{dx} = -2x \tan (x^2)

General Formulas and Concepts:

<u>Calculus</u>

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                 \displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle y = \ln (\cos x^2)

<u>Step 2: Differentiate</u>

  1. Logarithmic Differentiation [Derivative Rule - Chain Rule]:                       \displaystyle y' = \frac{(\cos x^2)'}{\cos x^2}
  2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:                   \displaystyle y' = \frac{-\sin x^2 (x^2)'}{\cos x^2}
  3. Basic Power Rule:                                                                                         \displaystyle y' = \frac{-2x \sin x^2}{\cos x^2}
  4. Rewrite [Trigonometric Identities]:                                                              \displaystyle y' = -2x \tan (x^2)

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

8 0
3 years ago
If the first of three consecutive integers is subtracted from 138, the result is the sum of the second and third. What are the i
otez555 [7]

Answer:

First Integer = n = 45

Second Integer = n+1 = 45 + 1 = 46

And Third Integer = n+ 2 = 45 +2 = 47

Step-by-step explanation:

Let First integer = n

Second Integer = n+1

Third Integer = n+2

According to the question given (If the first of three consecutive integers is subtracted from 138, the result is the sum of the second and third) the equation will be:

138 - n = (n+1) + (n+2)

Solving the equation:

138 - n = n+1+n+2

138 - n = 2n+3

138 - 3 =2n +n

135 = 3n

135/3 = n

=> n= 45

So, First Integer = n = 45

Second Integer = n+1 = 45 + 1 = 46

And Third Integer = n+ 2 = 45 +2 = 47

8 0
3 years ago
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