Towns A and B are located 6 miles and 3 miles, respectively, from a major expressway.

Question

The point on the expressway closest to town A is 12 miles from the point on the expressway closest to town B. Two new roads are to be built from A to the expressway and then to B. 

 

a. Express the combined lengths of the new roads in terms of x.

 

b. If the combined lengths of the new roads is 15 miles, what distance does x represent? 

Answer 1

This is the situation assuming the highway is straight, at least between two points closest to the towns, and :

                |
                | 3m
                |-----B ---
                |      ^
                | 12m
           6m | V
 A----------| ___
                |
                |

The problem is not clear about the two roads' locations, or the towns' locations relative to the highway (i.e. they could be on the same side of the highway) so I'm assuming that they form a straight line from A to B and that the towns are on opposite sides of the highway. Then:
The new roads together (from A to the highway and from the highway to B) form the hypoteneuse of a right triangle whose legs are ^ + 3 miles and 12 miles, respectively. Let x be the combined lengths of the roads. Then
x = sqrt((6+3)^2 + (12)^2) (Answer to part a)
= sqrt(81 + 144)
= sqrt(225)
= 15

Since this is the value specified in part b, the answer is that x represents the straight line distance between towns A and B.