Question

Help!(Optimization)... Two towns A&B are 13.0 miles apart and located 8.0 and 3.0 miles east of a long, straight highway. A construction company has a contract to build a road from town A to the highway and then to town B. Determine the length of the shortest road that meets these requirements.

Answer 1

the length of road = x+y
using pythagorean thm
\[x+y = \sqrt{a^2 +64} + \sqrt{(12-a)^2 +9}\]
to minimize x+y, set derivative equal to 0
\[\frac{a}{\sqrt{a^2 +64}} + \frac{a-12}{\sqrt{(12-a)^2 +9}} = 0\]
solving for a leads to a 4th degree polynomial
\[a^2 (a^2 -24a +153) = (12-a)^2 (a^2 +64)\]
find zero with graphing software
\[a = \frac{96}{11}\]
plug in to get min value for "x+y"
\[\min_{x+y} = \sqrt{265}\]