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heureka

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 #2
avatar+26397 
+15

Geometry Help

 

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

 

 

i = 7, b = 8,
A = i + b/2 − 1 = 10

 

see: https://en.wikipedia.org/wiki/Pick%27s_theorem

 

1. What is the area of this polygon?

 

i=33b=20A=33+2021A=42

 

 

2. What is the area of this polygon?

i=34b=21A=34+2121A=43.5

 

 

4. This figure is made up of a rectangle and parallelogram.
What is the area of this figure?

i=18b=18A=18+1821A=26

 

 

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10.02.2017
 #2
avatar+26397 
+5

Complex root:

 

x=1+ilnx=12ln(1+i)2lnx=ln(1+i)=ln(|1+i|)+iarg(1+i)||1+i|=12+12=2=ln(2)+iarg(1+i)|arg(1+i)=arctan(11)=π4=ln(2)+iπ42lnx=ln(2)+iπ4|:2lnx=12ln(2)+iπ8=ln(2)+iπ8lnx=ln(42)+iπ8x=eln(42)+iπ8=eln(42)eiπ8x=42eiπ8x1=42(cosπ8+isinπ8)cosπ8=122+2sinπ8=1222x1=42(122+2+i1222)x1=422+22+i42222x2=x1x2=422+22i42222

 

x11.09868+0.45509ix21.098680.45509i

 

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09.02.2017
 #2
avatar+26397 
+15

I am a number less than 3000.

If you divide me by 32 you will get a remainder 30.

If you divide me by 54 , you will get a remainder 44.

What number am I?

 

I am 638 or 1502 or 2366

 

n30(mod32)n44(mod54)or n30=32rn44=54s

 

Because 32r or 54s is even
so n-30 or n-44 is even so n is even


We can set: m=n2 and m is an integer.

 

n30=32r|:2n215=16rm15=16rn44=54s|:2n222=27sm22=27sor m15(mod16)m22(mod27) with n=2m and 1627=432

 

Because 16 and 27 are relatively prim ( gcd(16,27) = 1! ) we can go on:

m=1527[127(mod16)]+2216[116(mod27)]

 

[127(mod16)]=27φ(16)1(mod16)=2781mod16=277mod16=3[116(mod27)]=16φ(27)1(mod27)=16181mod27=1617mod27=22

 

m=15273+221622m=8959+k432|kZn=2mn=2(8959+k432)n=17918+k864nmin=17918(mod864)nmin=638n=638+k864|kZ

 

k=0:n=638k=1:n=638+1864n=1502k=2:n=638+2864n=2366

 

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09.02.2017
 #2
avatar+26397 
+20

Two pipes together drain a wastewater holding tank in 6 hours.
If used alone to empty the tank, one takes 2 hours longer than the other.
How long does each take to empty the tank when used alone?

 

Let q1=Volumetric flow pipe 1Let q2=Volumetric flow pipe 2Let V=Volume of the tank

 

q1t+q2t=V(q1+q2)t=V|t=6 hours(1)(q1+q2)6=V

 

(2)q1t1=Vt1=Vq1q2t2=V|t2=t1+2 hours(3)q2(t1+2)=Vq2(t1+2)=V|t1=Vq1q2(Vq1+2)=V(4)V=2q1q2q1q2

 

(1) = (4):

V=6(q1+q2)=2q1q2q1q26(q1+q2)=2q1q2q1q2|:23(q1+q2)=q1q2q1q2|(q1q2)3(q1+q2)(q1q2)=q1q23(q21q22)=q1q23q213q22q1q2=03q21q1q23q22=0q1=q2±q2243(3q22)23q1=q2±q22+36q226q1=q2±37q226q1=q2±q2376q1=16(1±37)q2q1=16(1+37)q2 or q1=16(137)q2 no solution q1<0!(5)q1q2=16(1+37)

 

(2) = (3):

V=q1t1=q2(t1+2)q1t1=q2(t1+2)(6)q1q2=t1+2t1

 

(5) = (6):

q1q2=16(1+37)=t1+2t116(1+37)=t1+2t1(1+37)t1=6(t1+2)(1+37)t1=6t1+12(1+376)t1=12(375)t1=12t1=12375t1=126.082762530305t1=121.08276253030t1=11.0827625303 hourst2=t1+2 hourst2=13.0827625303 hours

 

 

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08.02.2017
 #3
avatar+26397 
0

Find the sum of 35 terms of an arithmetic series
of which the first term is a and the fifteenth term is 9a.

 

Let t1 = a

Let t15 = 9a

 

Formula:

(1)tx=t1+(x1)d(2)ty=t1+(y1)d(3)tz=t1+(z1)dI=(1)(2)txty=d(xy)II=(1)(3)txtz=d(xz)II/Itxtztxty=d(xz)d(xy)txtztxty=xzxy|(txty)txtz=(txty)xzxytz=tx(txty)xzxytz=txtx(xzxy)+ty(xzxy)tz=tx(1xzxy)+ty(xzxy)tz=tx(xyx+zxy)+ty(xzxy)tz=tx(zyxy)+ty(xzxy)

 

t35 = ?

x=1tx=t1=ay=15ty=t15=9az=35tz=t35= ?tz=tx(zyxy)+ty(xzxy)t35=a(3515115)+9a(135115)t35=a(2014)+9a(3414)t35=a(107)+9a(177)t35=9a(177)a(107)t35=a(9177)a(107)t35=a(1537)a(107)t35=a(153107)t35=1437a

 

s35 = ?

s35=(t1+t352)35|t1=at35=1437as35=(a+1437a2)35s35=(1+14372)35as35=(15014)35as35=15052as35=755as35=375a

 

 

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08.02.2017