A motorboat travels 48 miles down a river in the same time it takes to travel 32 miles up the river. If the rate of the current is 4 miles per hour that is the rate of the boat still water?
If the speed is x, then since time = distance/speed,
48/(x+4) = 32/(x-4)
Let's assume the rate of the motorboat in still water is "b" miles per hour.
When the motorboat is traveling downstream (along with the current), the effective speed of the boat will be the sum of its speed in still water and the speed of the current, which is (b + 4) miles per hour.
Similarly, when the motorboat is traveling upstream (against the current), the effective speed of the boat will be the difference between its speed in still water and the speed of the current, which is (b - 4) miles per hour.
According to the given information, the motorboat takes the same time to travel 48 miles downstream as it does to travel 32 miles upstream. This can be expressed as:
Time taken downstream = Time taken upstream
Distance / Speed dowstream = Distance / Speed upstream
48 / (b + 4) = 32 / (b - 4)
To solve this equation, we can cross-multiply:
48(b - 4) = 32(b + 4)
48b - 192 = 32b + 128
48b - 32b = 128 + 192
16b = 320
b = 320 / 16
b = 20
Therefore, the rate of the boat in still water is 20 miles per hour.
To find the rate of the boat in still water, we can use the concept of relative speed.
Let's assume that the rate of the boat in still water is 'x' miles per hour. Since we know the rate of the current is 4 miles per hour, the effective speed of the boat when traveling downstream (along the current) would be (x + 4) miles per hour, and when traveling upstream (against the current) it would be (x - 4) miles per hour.
Now, we are given that the boat travels 48 miles downstream in the same time it takes to travel 32 miles upstream.
We can use the formula:
Time = Distance / Speed
Let's calculate the time it takes to travel 48 miles downstream:
Time downstream = 48 / (x + 4)
Now, let's calculate the time it takes to travel 32 miles upstream:
Time upstream = 32 / (x - 4)
Since we are given that these two times are equal, we can set up the following equation:
48 / (x + 4) = 32 / (x - 4)
To solve for x, let's cross-multiply and simplify the equation:
48(x - 4) = 32(x + 4)
48x - 192 = 32x + 128
48x - 32x = 128 + 192
16x = 320
x = 320 / 16
x = 20
Therefore, the rate of the boat in still water is 20 miles per hour.