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Brums [2.3K]
2 years ago
5

The ten sides of a regular decagon are colored with five different colors, so that all five colors are used, and sides that are

diametrically opposite have the same color. One possible coloring is shown below.
How many different ways are there to color the sides of the decagon? (Two colorings are considered identical if one can be rotated to form the other.)

Mathematics
1 answer:
const2013 [10]2 years ago
6 0

Answer:

  • <u>120</u>

Explanation:

Number the sides of the decagon: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, from top (currently red) clockwise.

  • The side number one can be colored of five different colors (red, orange, blue, green, or yellow): 5
  • The side number two can be colored with four different colors: 4
  • The side number three can be colored with three different colors: 3
  • The side number four can be colored with two different colors: 2
  • The side number five can be colored with the only color left: 1
  • Each of the sides six through ten can be colored with one color, the same as its opposite side: 1

Thus, by the multiplication or fundamental principle of counting, the number of different ways to color the decagon will be:

  • 5 × 4 × 3 × 2 ×1 × 1 × 1 × 1 × 1 × 1 = 120.

Notice that numbering the sides starting from other than the top side is a rotation of the decagon, which would lead to identical coloring decagons, not adding a new way to the number of ways to color the sides of the figure.

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The balcony of an apartment is 4 feet by 7 feet. On a scale drawing of the apartment, the balcony is 0.8 inch by 1.4 inches, and
-Dominant- [34]

Answer:

The actual dimensions of the kitchen are 12\ ft by 14\ ft

Step-by-step explanation:

step 1

Find the scale  of the drawing

To find the scale , remember that is equal to the ratio of  the corresponding dimension in the drawing divided by the corresponding actual dimension

so

\frac{0.8}{4}\frac{in}{ft}=0.2\frac{in}{ft}

step 2

To find the actual dimensions of the kitchen, divide the dimension in the drawing by the scale

so

2.4\ in=2.4/0.2=12\ ft

2.8\ in=2.8/0.2=14\ ft

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3 years ago
How many quarts of soil are used to fill 2 flower pots? Each pot holds 3 2/5
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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

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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.

5 0
2 years ago
Simplify the expression:<br> (1 2/7)(-4 1/5)
stealth61 [152]

Answer:

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Step-by-step explanation:

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1 2/7 to 9/7

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Answer:

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Hence, correct answer is D.

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