A boat traveling downstream a distance of 39 miles then came right back.If the speed of the current was 12 mph and the total trip took 4 hours 20 minutes,find the average speed of the boat relative to the water.
speed of boat --- x mph
time to go downstream = 39/(x+12)
time to go upstream = 39/(x-12) , clearly x > 12
39/(x+12) + 39/(x-12) = 4 1/3 = 13/3
(39(x-12) + 39(x+12))/(x^ - 144) = 13/3
78x/(x^2 - 144) = 13/3
13x^2 - 1872 = 234x
13x^2 - 243x - 1872 = 0
x^2 - 18x - 144 = 0
completing the square, (faster than the formula for this one)
x^2 - 18x + 81 = 144 + 81
(x-9)^2 = 225
x-9 =±15
x = 9 ±15 = 24 or an inadmissable -6
the boat's speed was 24 mph
check:
39/36 + 39/12
= 156/36
= 13/3 or 4 1/3 or 4 hrs 20 minutes
what the hell
To find the average speed of the boat relative to the water, you need to calculate the average of the boat's speed while traveling downstream and upstream.
Let's start by finding the time it took for the boat to travel downstream.
The total trip took 4 hours 20 minutes, which can be converted to hours by dividing by 60:
4 hours + 20 minutes/60 = 4 hours + 1/3 hour = 4(3/3) + 1/3 = 13/3 hours.
Since the boat traveled downstream and then back, the time spent traveling downstream will be half of this total time: (13/3) / 2 = 13/6 hours.
The speed of the current is given as 12 mph. This means that while traveling downstream, the boat's speed relative to the water is increased by 12 mph.
Let's represent the average speed of the boat relative to the water as x mph.
To find the boat's speed downstream, we need to add the speed of the current: x + 12 mph.
Now, let's calculate the distance traveled downstream. Since distance = speed * time, the distance traveled downstream is (x + 12) * (13/6).
Now, let's find the time taken to travel upstream. The time taken to travel upstream will be the total time minus the time taken to travel downstream:
Total time - Time taken downstream = (13/3) - (13/6) = 13/6 hours.
When the boat is traveling upstream, the speed relative to the water is reduced by the speed of the current. Thus, the speed of the boat upstream is x - 12 mph.
Considering the distance traveled upstream is the same as the distance traveled downstream (39 miles), we can set up an equation:
(x - 12) * (13/6) = 39
Now, let's solve the equation for x:
13(x - 12) = 6 * 39
13x - 156 = 234
13x = 234 + 156
13x = 390
x = 390/13
x = 30
Therefore, the average speed of the boat relative to the water is 30 mph.
To find the average speed of the boat relative to the water, we need to consider the effects of the current. Let's break down the problem step by step:
1. Let's assume the speed of the boat in still water is 'x' mph, and the speed of the current is given as 12 mph.
2. When the boat is traveling downstream, it gets an additional speed boost from the current. So, the effective speed of the boat relative to the ground (downstream) would be the sum of the boat's speed in still water and the speed of the current. Therefore, the speed downstream would be (x + 12) mph.
3. When the boat is traveling upstream, it has to overcome the opposing force of the current, which reduces its effective speed. So, the effective speed of the boat relative to the ground (upstream) would be the difference between the boat's speed in still water and the speed of the current. Therefore, the speed upstream would be (x - 12) mph.
4. The total distance covered by the boat is 39 miles downstream and then another 39 miles upstream, resulting in a total distance of 78 miles.
5. The total time taken for the round trip is given as 4 hours 20 minutes, which is equivalent to 4.33 hours.
Now, let's calculate the average speed of the boat relative to the water:
Average speed = Total distance / Total time
Since the total distance is 78 miles and the total time is 4.33 hours:
Average speed = 78 miles / 4.33 hours
Calculating this gives us:
Average speed = 18.03 mph
Therefore, the average speed of the boat relative to the water is 18.03 mph.