A boat travels 8km upstream and back in 2 hours. If the current flows at a constant speed of 3km/h, Find the speed of the boat in still water.(when the boat goes upstream, it speed reduces by 3km/h, and when going downstream, its speed increases by 3 km/h)
Let the speed of the boat in still water be x km/h
time upstream = 8/(x-3)
time downstream = 8/(x+3)
8/(x-3) + 8/(x+3) = 2
times (x+3)(x-3)
8(x+3) + 8(x-3) = 2(x+3)(x-3)
8x + 24 + 8x - 24 = 2x^2 - 18
2x^2 -16x - 18 = 0
x^2 - 8x - 9 = 0
(x-9)(x+1) = 0
x = 9 or x = -1, a negative x is not likely
x = 9
check:
speed upstream = 6 km/h, time = 8/6 or 4/3 hrs
speed downstrem = 12 km/h, time = 8/12 = 2/3 hr
total time = 4/3 + 1/3 = 2
all is good!
To find the speed of the boat in still water, we can use the formula:
Speed of boat in still water = (Speed downstream + Speed upstream) / 2
Let's break down the problem step by step:
1. Let's assume the speed of the boat in still water is x km/h.
2. When the boat goes downstream, its speed (relative to the shore) is x + 3 km/h because the current adds an additional speed of 3 km/h.
3. When the boat goes upstream, its speed (relative to the shore) is x - 3 km/h because the current subtracts a speed of 3 km/h.
4. Now, let's calculate the time it takes for the boat to travel 8 km downstream.
Time downstream = Distance / Speed downstream
Time downstream = 8 km / (x + 3) km/h
Time downstream = 8 / (x + 3) hours
5. Similarly, let's calculate the time it takes for the boat to travel 8 km upstream.
Time upstream = Distance / Speed upstream
Time upstream = 8 km / (x - 3) km/h
Time upstream = 8 / (x - 3) hours
6. According to the problem, the total time it takes for the boat to travel 8 km upstream and back is 2 hours.
Total time = Time downstream + Time upstream
2 hours = 8 / (x + 3) + 8 / (x - 3)
7. Now let's solve this equation to find the value of x, which represents the speed of the boat in still water.
Multiply both sides of the equation by (x + 3)(x - 3):
2(x + 3)(x - 3) = 8(x - 3) + 8(x + 3)
8. Simplify and solve for x:
2(x^2 - 9) = 8x - 24 + 8x + 24
2x^2 - 18 = 16x
Rearrange the equation:
2x^2 - 16x - 18 = 0
9. Now we have a quadratic equation. We can either factor, complete the square, or use the quadratic formula to solve for x. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values: a = 2, b = -16, c = -18
x = (-(-16) ± √((-16)^2 - 4 * 2 * -18)) / (2 * 2)
x = (16 ± √(256 + 144)) / 4
x = (16 ± √400) / 4
x = (16 ± 20) / 4
We get two possible values for x:
x = (16 + 20) / 4 = 9 km/h
x = (16 - 20) / 4 = -1 km/h
Since the speed cannot be negative, the speed of the boat in still water is 9 km/h.
Therefore, the speed of the boat in still water is 9 km/h.