The speed of a stream is 6 mph. If a boat travels 52 miles downstream in the same time that it takes to travel 26 miles upstream, what is the speed of the boat in still water?
let the speed of the boat in still water be x mph
using Distance = rate x time or Time = distance/rate
upstream:
time = 26/(x-6)
downstream:
time = 52/(x+6)
26/(x-6) = 52/(x+6)
cross-multiply
52x - 312 = 26x + 156
26x = 468
x = 18
boat's speed is 18 mph
To find the speed of the boat in still water, we need to break down the boat's motion into its two components: the speed of the boat itself and the speed of the current.
Let's assume that the speed of the boat in still water is represented by "B" (in mph), and the speed of the current is represented by "C" (in mph).
When the boat is moving downstream, it is adding its own speed to the current speed, so its effective speed will be B + C. Similarly, when the boat is moving upstream, it is subtracting the current speed from its own speed, so the effective speed will be B - C.
We are given that the speed of the stream (current) is 6 mph.
Now, let's use the formula
time = distance / speed
For the downstream trip, we can write the equation as:
time downstream = distance downstream / (boat speed + current speed)
Given that the boat travels 52 miles downstream, we have:
time downstream = 52 / (B + 6)
For the upstream trip, we can write the equation as:
time upstream = distance upstream / (boat speed - current speed)
Given that the boat travels 26 miles upstream, we have:
time upstream = 26 / (B - 6)
According to the problem, the time taken for both trips is the same. Mathematically, we can write this as:
time downstream = time upstream
Therefore:
52 / (B + 6) = 26 / (B - 6)
To solve this equation and find the value of B, we can cross-multiply:
52 * (B - 6) = 26 * (B + 6)
Expanding:
52B - 312 = 26B + 156
Simplifying:
52B - 26B = 156 + 312
26B = 468
Dividing by 26:
B = 18
The speed of the boat in still water is 18 mph.