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1 answer:
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Answer:
a) Malvern - Sheridan - Pine Bluff
b) Camden - Fordyce - Pine Bluff
c) Benton - Malvern - Sharidan - Arkadepia
d) Benton - Malvern - Sharidan - Arkadepia - Gurden - Camden - Fordyce
e) Benton - Malvern - Sharidan - Arkadepia - Gurden - Camden
Step-by-step explanation:
P.S - The exact question is -
Given - According to a map of Arkansas, a number of good roads connect
some of the towns south of Little Rock. The road between
Malvern and Benton is 23 miles long, and between Benton and Pine
Bluff is 62 miles long. The road between Malvern and Sheridan is
28 miles long, and then from Sheridan to Pine Bluff is 22 miles
long. Arkadelphia is connected to two towns: it is 12 miles from
Gurden and 21 miles from Sheridan. Fordyce has three roads to
these towns: 45 miles to Pine Bluff, 39 miles to Sheridan, and 27
miles to Camden. The road from Gurden to Camden is 38 miles.
To find - Unfortunately, the road from Fordyce to Sheridan is closed due to
dangerous road conditions caused by the recent bad weather. Find
the shortest route between each pair of towns:
a) Malvern and Pine Bluff
b) Camden and Pine Bluff
c) Benton and Arkadelphia
d) Fordyce and Malvern
e) Benton and Camden
Proof -
The figure is as follows :
a)
Shortest route between Malvern and Pine Bluff :
Firstly, put all the vertices to ∞
Start from Malvern ,
Check the shortest distance from the corresponding vertices of Malvern.
Shortest Distance = min { Malvern - Benton, Malvern -Sheridan }
= min { 23, 28 } = 23
⇒Shortest distance is 23 ( Malvern -Benton )
Now,
Check shortest distance from Benton and Malvern
Shortest Distance = min { Benton - Pine Bluff, Malvern -Sheridan }
= min { 62, 28 } = 28
⇒Shortest distance is 28 ( Malvern -Sheridan )
Now,
Check shortest distance from Benton and Sheridan
Shortest Distance = min { Benton - Pine Bluff, Sheridan - Pine-Bluff}
= min { 85, 50 } = 50
⇒Shortest distance is 50 ( Sheridan - Pine-Bluff )
So,
By back tracking , we get
Shortest Route be -
Malvern - Sheridan - Pine Bluff
And the Shortest distance be = 50 miles
b)
Shortest Route between Camden and Pine Bluff
From Camden -
Shortest Distance = min {Camden-Fordyce, Camden-Gurden }
= min { 27, 38 } = 27
⇒Shortest Distance = 27 (Camden-Fordyce )
Now,
Check Shortest distance from Camden and Fordyce
Shortest Distance = min { Camden-Gurden, Fordyce-Pine Bluff }
= min { 38, 72 } = 38
⇒Shortest Distance = 38 ( Camden-Gurden )
Now,
Check Shortest Distance from Gurden and Fordyce
Shortest Distance = min { Gurden-Arkadepia, Fordyce-Pine Bluff }
= min { 50, 72 } = 50
⇒Shortest Distance = 50 ( Gurden-Arkadepia )
Now,
Check Shortest Distance from Arkadepia and Fordyce
Shortest Distance = min { Arkadepia - Sharidan, Fordyce-Pine Bluff }
= min { 71, 72 } = 71
⇒Shortest Distance = 71 ( Arkadepia - Sharidan )
Now,
Check Shortest Distance from Sharidan and Fordyce
Shortest Distance = min { Sharidan - Pine Bluff , Fordyce-Pine Bluff }
= min { 93, 72 } = 72
⇒Shortest Distance = 72 ( Fordyce-Pine Bluff )
So, By Back tracking , we get
Shortest route be
Camden - Fordyce - Pine Bluff
Shortest Distance be 72 miles
c)
Shortest route between Benton and Arkadelphia
By applying the same method ,
If we start from Benton and by using Pine bluff lane, distance be 62 + 45 + 27 + 38 + 12 = 184 miles
And
If we start from Benton and by using Malvern lane, distance be 23 + 28 + 21 = 72 miles
So,
Shortest Route be -
Benton - Malvern - Sharidan - Arkadepia
Shortest distance be 72 miles
d)
Shortest route between Fordyce and Malvern
By applying the same method ,
If we start from Malvern and by using Benton lane, distance be 23 + 62 + 45 = 130 miles
And
If we start from Malvern and by using Sharidan lane, distance be 28 + 21 + 12 + 38 + 27 = 126 miles
So,
Shortest Route be -
Benton - Malvern - Sharidan - Arkadepia - Gurden - Camden - Fordyce
Shortest distance be 126 miles
e)
Shortest route between Benton and Camden
By applying the same method ,
If we start from Benton and by using Pine Bluff lane, distance be 62 + 45 + 27 = 134 miles
And
If we start from Benton and by using Malvern lane, distance be 23 + 28 + 21 + 12 + 38 = 122 miles
So,
Shortest Route be -
Benton - Malvern - Sharidan - Arkadepia - Gurden - Camden
Shortest distance be 122 miles
