A boat has a 10 gallon gasoline tank and travels at 20mi/hr with a fuel consumption of 16mi/gal when operated at full throttle in still water. The boat is moving upstream into a 5mi/hr current. How far upstream can the boat travel and return on 10 gallons of gasolineif it is operated at full throttle during the entire trip?
consumption rate against current = (15 mile/hour)/(16 miles/gallon)
= 15 gallons/16 hours
= (15/16) gallons/hour
consumption rate with the current = (25 miles/hour)/(16 miles/gallon)
= (25/16) gallons/hour
let t1 hours be the time going against the current
distance = 15t1
let t2 hours be the time going with the current
distance = 25t2
but the went the same distance, so
15t1 = 25t2
t1 = (25/15)t2 = 5t2/3
Time x consumption rate = consumption
t1hrs(15/16) gallons/hrs + t2hrs(25/16)gallons/hr = 10 gallons
t1(15/16) + t2(25/16) = 10
times 16
15t1 + 25t2 = 160
but 15t1 = 25t2
50t2 = 160
t2 = 3.2 hrs = 3 hrs, 12 minutes
t1 = 5.3333 hrs. = 5 hrs. 20 minutes
distance = 25t2 = 80 miles
check: 15(5.33333333...) = 80 miles
So the boat can go 80 miles upstream and come back the same 80 miles.
(Wow, did I just prove that to go 160 miles at a rate of 16 miles/gall would take 10 gallons???)
To determine how far upstream the boat can travel and return on 10 gallons of gasoline, we need to consider the effect of the current and calculate the overall speed of the boat.
First, let's determine the boat's speed relative to the still water. Since it travels at 20 mi/hr in still water and there is a 5 mi/hr current, the boat's speed relative to the still water would be 20 - 5 = 15 mi/hr.
Next, we need to find the fuel consumption rate of the boat while moving with the current. Since the boat's fuel consumption rate is 16 mi/gal in still water, it will also consume 16 mi/gal when moving with the current.
Now, we can calculate how far the boat can travel upstream. Since the boat's fuel consumption rate is 16 mi/gal and it has a 10-gallon gasoline tank, it can travel 16 * 10 = 160 miles upstream.
To determine the total distance the boat can cover (including the return trip), we need to consider the effect of the current. When the boat is moving upstream, it travels at a speed of 15 mi/hr. However, when returning downstream with the current, the boat's speed is increased by the current speed. Therefore, the boat's speed downstream would be 15 + 5 = 20 mi/hr.
Since the boat will consume fuel at a rate of 16 mi/gal while moving downstream, the total distance the boat can cover (including the return trip) on 10 gallons of gasoline would be:
Distance upstream + Distance downstream = 160 miles + (16 mi/gal * 10 gal) = 160 + 160 = 320 miles.
Therefore, the boat can travel a distance of 320 miles upstream and back on 10 gallons of gasoline if operated at full throttle throughout the entire trip.