Problema Solution
A military hovercraft (a vehicle that can travel on land and over water) is stationed 50 miles directly off the coast from an abandoned bunker. 120 miles down the shoreline from the bunker, there is a group of troops that need to be picked up by the hovercraft. The hovercraft uses 6 gallons of fuel per mile while traveling over water and 3 gallons of fuel per mile while traveling over land. If the hovercraft has a total of 650 gallons of fuel remaining, how far down the beach from the bunker should the hovercraft land in order to pick up the troops? Your answer must be accurate to the nearest foot.
Answer provided by our tutors
the hovercraft, bunker and troop location form a right triangle whose sides are 50, 120 and sqrt(50^2 + 120^2) = 130
we need to define another right triangle whose side is 'x' such that '120-x' represents the distance travelled by land and sqrt(50^2 + x^2) represents the distance travelled by water
we need to pick up and retrieve the troops using the 650 gallons, therefore we only want to use 325 gallons picking up the troops, otherwise we simply strand ourselves together somewhere, so we base our distances on 325 gallons of fuel
let 'g' represent the gallons of fuel used to travel by land, then '325-g' represents the gallons of fuel used to travel by water
distance = mpg * g
g = distance / mpg
we travel 120-x miles using 'g' gallons at 3mpg:
120-x = 3g
we travel sqrt(50^2 + x^2) miles using '325-g' gallons at 6mpg:
sqrt(50^2 + x^2) = 6(325-g)
120-x = 3g, so g = (120-x)/3
sqrt(50^2 + x^2) = 6(325-(120-x)/3)
solving for 'x' we have x=-569.269, so something is not right or the question should specify that we might be taking a pretty silly route by directing the hoover craft quite a distance in the wrong direction before driving along the shoreline
the longest route would be 50 miles by water, 100 miles by land and then return, with a total fuel requirement of:
2(50/6 + 100/3) = 83 1/3 gallons
the longest water route would be (sqrt(50^2 + 120^2) = 130 miles), with a total fuel requirement of:
(130*2)/6 = 43 1/3 gallons
...so, in the end, there is simply -plenty of fuel to land anywhere they want to on the beach - pick some spot to the nearest foot and it will work! ...or convert 569.269 miles to feet and land the craft way down shore the opposite direction before continuing on land