An unmanned aerial vehicle (UAV) carries 45 gallons of fuel. It burns fuel at a rate of 3 gph (gallons/hour) and flies at a speed of 160 mph (miles/hour). We want to fly it to a location X, fly around the area for 1 hour, and then fly it back to the launching point.
How far away can location X be if we want the UAV to arrive back home with a fuel reserve of 0.5 hour?
The greatest number of miles that location x can be is 1120 miles
We will use up 3 gallons flying in a circle for an hour and we want .5*3 = 1.5 gallons left over so we can use
g = 45 - 3 - 1.5 gallons
g gallons * 1 hr/3 gal = x hours to fly out and back
that is x/2 hours one way
so distance = (x/2)*160 miles
g = 40.5 gallons to fly with
40.5/3 = 13.5 hours out and back
so 6.75 hours out
6.75 * 160 = distance out
To calculate the maximum distance the UAV can fly to location X and still have a fuel reserve of 0.5 hours, we need to consider the fuel consumption during the round trip and the flight time for the round trip.
First, let's calculate the fuel consumption for the round trip:
Fuel burned per hour = 3 gallons/hour
Flight time for the round trip = 2 hours (to location X and back)
Fuel burned for the round trip = Fuel burned per hour x Flight time for the round trip = 3 gallons/hour x 2 hours = 6 gallons
Now, let's calculate the remaining fuel after the round trip:
Initial fuel = 45 gallons
Fuel burned for the round trip = 6 gallons
Remaining fuel after the round trip = Initial fuel - Fuel burned for the round trip = 45 gallons - 6 gallons = 39 gallons
Next, let's calculate the flight time for the outbound journey and the return journey:
Outbound flight time = Distance / Speed
Return flight time = Distance / Speed
We want the UAV to have a fuel reserve of 0.5 hours after the round trip. Therefore, the total flight time for the round trip should be 2 hours + 0.5 hours = 2.5 hours.
Knowing that the flight time is the distance divided by the speed, we can set up the following equation:
Outbound flight time + Return flight time = 2.5 hours
(Distance / Speed) + (Distance / Speed) = 2.5 hours
2 * Distance / Speed = 2.5 hours
Distance / Speed = 2.5 hours / 2
Distance = Speed * (2.5 hours / 2)
Distance = 160 mph * (2.5 hours / 2)
Distance = 200 miles
Therefore, the maximum distance to location X can be 200 miles while still ensuring a fuel reserve of 0.5 hours.