The current of a river moves at 3 miles per hour. It takes a boat 3 hours to travel 12 miles upstream, against the current, and return the same distance traveling downstream, with the current. What is the boats rate in still water?
boat's speed in still water --- x mph
against the current ----- x-3 mph
with the current -------- x + 3 mph
time against current = 12/(x-3)
time with the current = 12/(x+3)
12/(x+3) + 12/(x-3) = 3
times (x-3)(x+3) or x^2 - 9
12(x-3) + 12(x+3) = 3(x^2 - 9)
12x-36 + 12x + 36 = 3x^2 - 27
3x^2 - 24x - 27 = 0
x^2 - 8x - 9 = 0
(x-9)(x+1) = 0
x = 9 or x = -1, I guess we'll reject that negative
the boat's speed in calm water is 9 mph
check:
12/12 + 12/6
= 1 + 2
= 3hrs.
Well, let me calculate it while adding some humor to the equation. With the current, the boat goes downstream at a speed of 3 miles per hour. And against the current, it takes 3 hours to travel 12 miles upstream. So, we can say that the current is definitely good at creating obstacles! Now, to find the boat's rate in still water, let's call it "B".
When the boat goes upstream, it's like a snail fighting against a strong breeze. So, the effective speed of the boat would be the boat's rate in still water (B) minus the current speed (3 miles per hour).
And when the boat goes downstream, it's like a cheetah on a water slide. So, the effective speed would be the boat's rate in still water (B) plus the current speed (3 miles per hour).
Since the distance covered is the same in both directions (12 miles), we can set up an equation:
12/(B - 3) = 12/(B + 3)
By solving this equation, we find that the boat's rate in still water (B) is 6 miles per hour. So, even though the current may cause some trouble, the boat can zoom along at a decent speed in still water!
To find the boat's rate in still water, we can use the formula:
Boat's rate in still water = (Downstream speed + Upstream speed) / 2
Let's first find the upstream and downstream speeds:
Upstream speed = Boat's rate in still water - Current speed
Downstream speed = Boat's rate in still water + Current speed
Given that the current of the river moves at 3 miles per hour, we can substitute this in the equations:
Upstream speed = Boat's rate in still water - 3
Downstream speed = Boat's rate in still water + 3
Now, we are given that it takes the boat 3 hours to travel 12 miles upstream and the same distance downstream. We can use the equation:
Time = Distance / Speed
For the upstream travel:
Time = 3 hours
Distance = 12 miles
Speed = Upstream speed = Boat's rate in still water - 3
Rearranging the equation, we have:
Upstream speed = Distance / Time = 12 miles / 3 hours = 4 miles per hour
Similarly, for the downstream travel:
Time = 3 hours
Distance = 12 miles
Speed = Downstream speed = Boat's rate in still water + 3
Again, rearranging the equation, we get:
Downstream speed = Distance / Time = 12 miles / 3 hours = 4 miles per hour
Now, we can write the equations we derived:
Upstream speed = Boat's rate in still water - 3 = 4 miles per hour
Downstream speed = Boat's rate in still water + 3 = 4 miles per hour
Solving these equations simultaneously, we get:
Boat's rate in still water = 4 miles per hour + 3 miles per hour = 7 miles per hour
Therefore, the boat's rate in still water is 7 miles per hour.
To find the boat's rate in still water, we need to understand the effect of the river's current on the boat's speed.
Let's assume the boat's rate in still water is represented by "B" (in miles per hour), and the speed of the river's current is represented by "C" (in miles per hour).
When the boat is traveling upstream against the current, its effective speed is reduced by the speed of the current. So, the boat's speed becomes (B - C) miles per hour.
Similarly, when the boat is traveling downstream with the current, its effective speed is increased by the speed of the current. So, the boat's speed becomes (B + C) miles per hour.
Now, let's apply this knowledge to the given scenario:
Upstream journey:
- Rate of the boat (B - C) = 12 miles / 3 hours = 4 miles per hour
Downstream journey:
- Rate of the boat (B + C) = 12 miles / 3 hours = 4 miles per hour
We can set up a system of equations to solve for B and C:
B - C = 4 (Equation 1)
B + C = 4 (Equation 2)
Adding these two equations together eliminates C, giving:
2B = 8
Simplifying, we find that B = 4 miles per hour.
Therefore, the boat's rate in still water is 4 miles per hour.