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习题练习:IB MAI HL Geometry & Trigonometry Topic 3.3 Voronoi Diagrams



 作者: admin   总分: 12分  得分: _____________

答题人: 匿名未登录  开始时间: 24年02月08日 21:21  切换到: 整卷模式

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1#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Cellular phone towers in a town are represented by the points P, f:4b2;hu vaheQ, R, and S (not shown) in the partially complete Voronoi diagram shown below. The vhh u;4abf:e2 regions created by the diagram represent the phone coverage provided by each tower.



1.Write down the coordinates of the missing phone tower S. (a,b) a=   b=  
2.Calculate the coverage area provided by phone tower Q. A =    $km^2$
3.Find the gradient of the edge of Voronoi diagram between towers Q and R. $m_{e,QR}$ =   
The phone towers provide different data download speeds, as per the table below.


Tower is P ;Data download speed is 25 MBps ;Tower is R ;Data download speed is 30 MBps ;Tower is Q & S ;Data download speed is 20 MBps ;
4.A customer is located at (4,10). Determine the download speed she receives.

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2#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the Voronoi diagram below with sinqarst( f- 9t60b mtm6tes A, B, C and D, and intersections E and F.amt(fn9t06 qr ms- tb6



1.Determine the site closest to the coordinate (9,6).
2.Determine the coordinates of the centre of the largest empty circle.
3.Calculate the area of the largest empty circle. a ≈    $units^2$

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3#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Points A(1,5),B(5,9),C(7,7),cp6i/c.byqsl y q+v*1D(9,3) and E(13,11) represent feeding-racks for dears in the Teutoburg ylscq cv+b./yp6iq1* Forest in Germany. The feeding-racks are illustrated in the following coordinate axes.
Horizontal scale: 1 unit represents 1 km.
Vertical scale: 1 unit represents 1 km.

1.Calculate the gradient of the line segment CE.
Three straight lines are drawn on the coordinate axes to form an incomplete Voronoi diagram. $m_{CE}$ = $\frac{a}{b}$ a =    b =   
2.Find the equation of the line that would complete the Voronoi cell containing site C. Give your answer in the form
ax+by+d=0 where a,b,d∈Z. ax+by+c=0 a =    b =    c =   

3.In the context of the question, explain the meaning of the Voronoi cell containing site C.

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4#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Points A(−3,5), B(5,3), C(3,9), D(3,−3) and E(11,5) represent coff/2ofl ex vxh357dyk4 yee shops in a city. The coffee shops are illustrated on the following coordinate axes. y x5e 32olx7vyh4 f/dk
Horizontal scale: 1 unit represents 1 km.
Vertical scale: 1 unit represents 1 km.



1.Calculate the gradient of the line segment BE.
The diagram below shows three straight lines that form an incomplete Voronoi diagram.



2.Find the equation of the line completing the Voronoi cell containing site B. Give your answer in the form ax+by+d=0 where a,b,d∈Z. ax+by+c=0 a =    b =    c =   
3.In the context of the question, explain the meaning of the Voronoi cell containing site B.

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5#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The Voronoi diagram below shows four zw2rgbh *:0osfc6x* ; ncetd8hotels in a small town represented by points with coordinates A(−4,4), B(3,5), C(3,−3), and D(−1,3). The vert*8c n;rd :xzcb0fe2g *o6hts wices $V_1$​, $V_2$​ and $V_3$​ are also shown. Distances in the direction of the x and y axes are measured in increments of 100 metres.


1.Find the midpoint of AD. M (a,b) a=   b=  
2.Hence, find the equation of the line that passes through $V_1$ and$V_2$​. y =ax+b a =    b =   
The equation of line that passes through $V_1$​ and $V_3$​ is y=−2x+6.
3.Find the coordinates of $V_1$.(a,b) a=   b=  
The coordinates of $V_2$​ are (−5,−4) and the coordinates of V3V3​ are (2.5,1).
4.Find the distance from $V_1$​ to $V_2$​. Give your answer to the nearest metre. D ≈   
5.Given that the distance from $V_1$ to $V_3$ is 783 metres, find the angle $V_2\widehat{V_1}V_3​$​. Give your answer to the nearest degree. $V_2\widehat{V_1}V_3​$​ ≈    $^{\circ}C$
6.Hence, find the area of the Voronoi cell containing hotel D, giving your answer in $m^2$, to three significant figures. The manager of hotel D believes that the larger the area of triangle $V_1V_2V_3​$, the more people will stay at hotel D.A ≈    $m^2$
7.State one criticism of the manager's belief.


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6#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the Voronoi diagram below for a city that contains the towns A, B, C, np jjrtyt,h()db,82thk zk p 3 :xou93qov(s9andnt32p,dops vok8bk xu9t9)j3zh t( jrh: (yq, D.
The equation of the perpendicular bisector between sites A and D, PB(A,D), is found to be
y=x−25​

1.Determine the equation of the perpendicular bisector between C and D, PB(C,D).y =-$\frac{a}{b}$x+$\frac{c}{d}$ a =    b =    c =    d =   
2.Hence, determine the coordinates of the intersection $V_2$​. (a,b) a=   b=  
3.Determine the optimal position for a toxic waste dump in this city, such that it is as far away from a town as possible.

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7#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Consider the Voronoi diagram below that shows the location zolilp+2l96au5 ;)j z n;jpxk of convenient stores within a town. The convenient stores are labelled;i z2 ;ka+p 65 l9jzolp)njuxl A, B, C, and D.
The equation of the perpendicular bisector between sites A and D, PB(A,D), is found to be
y=−x+6​






1.Determine the equation of the perpendicular bisector between sites C and D, PB(C,D).y =$\frac{a}{b}$x-$\frac{c}{d}$ a =    b =    c =    d =   
2.Hence, determine the coordinates of the intersection $V_1$​.
A company that opens and manages convenient stores wants to open a new store in this town. (\frac{a}{b},\frac{c}{d}) a =    b =    c =    d =   
3.Determine the optimal position for this new store, so that it is as far as possible from its nearest store.

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8#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A town currently has four Italian pizza restaurants. A famous chef wishes to8wmitm:c.b;n 6rwh3a:ltqdv8d v1o /b jy 19y open a new pizza restaurant in the town but wants it to be located as far away as possible from the four cu8 89q:h y 1mcowtim 6d/yl3jbb1 r.tvavd:wn;rrent restaurants.
The Voronoi diagram below showing the positions of the four current restaurants on a set of coordinates axes: A(2,3) B(8,4), C(7,1), and D(2,1), where one unit represents 1 km.



1.Write down the coordinates of P. (a,b) a=   b=  
2.Show that the coordinates of Q are(5.03,3.32), correct to three significant figures.(a,b) a=   b=  
3.Determine where the famous chef should position their new restaurant.

4.Determine the estimated average weekly number of pizzas this new shop will sell. ≈    pizzas

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9#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The Voronoi diagram below shows the location of fire stations in outback8rk:2 6fa7up6r: u1zrfw ffa-t v9c wo Queensland, a North-Eastern state of A:8u-a1tp2c7a u6 :kvf 9r rwf6fzwfroustralia.
Horizontal scale: 1 unit represents 100 km.
Vertical scale: 1 unit represents 100 km.



1.Identify the nearest fire-station to houses located at
(1)(300,300)   
(2)(−200,100)   
A house located at (200,200) has an internal fire and requires assistance.
2.Determine which fire-stations the house is closest to.    and   
3.Determine how far the house is from these two fire-stations. Give your answer to the nearest kilometre. d ≈    km

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10#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The Voronoi diagram below shows the location of fire stawb:-d,i tzy(w q4 i -m5zmtrn2tions in outback Queensland,-bwn,:(-z yq tm5td2iwr4 imz a North-Eastern state of Australia.
Horizontal scale: 1 unit represents 100 km.
Vertical scale: 1 unit represents 100 km.



1.Identify the nearest fire-station to houses located at
(1)(300,300)   
(2)(−200,100)   
A house located at (200,200) has an internal fire and requires assistance.
2.Determine which fire-stations the house is closest to.    and   
3.Determine how far the house is from these two fire-stations. Give your answer to the nearest kilometre. d ≈    km

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11#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The Voronoi diagram below shows the fitne csbv(,wv,m io1*ko3hss studios A, B and C, located in a particulahoo*v k 1iw (b,3,csmvr district.



1.On the diagram, shade the cell with the missing fitness studio.
2.Determine the coordinates of the missing fitness studio, D.
3.Jennifer lives an equal distance from fitness studios A and D. Find the shortest possible distance Jennifer's house could be from fitness studios A and D. d ≈    km

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12#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The sites A, B (not shown), dtgk 4us 7k5a6 of*l,pC, and D in the Voronoi diagram below represent the location of rainfall ,* f6ok dg7tls4ua5kprecording gauges on a large corn farm in Texas, USA.
Horizontal scale: 1 unit represents 1 km.
Vertical scale: 1 unit represents 1 km.

1.Site B is missing from the diagram. Write down its coordinates. B : (a,b) a=   b=  
2.Calculate the area of the cell C. A =    $km^2$
3.Find the distance from point E(9,8) to the nearest rainfall recording gauge.
The table below shows measurements of the average monthly rainfall (mm) of each of the recording gauges. d =    km

4,Estimate the average monthly rainfall at point F(1,6) in mm.    mm

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