A boat travels 1 km upstream and 1 km back. The time for the round trip is 4 hours. The speed of the stream is 5 km/hr. What is the speed of the boat in still water?

let speed of boat be x mph

time upstream = 1/(x-5)
time downstream = 1/(x+5)

1/(x-5) + 1/(x+5) = 4
multiply each term by (x+5)(x-5)

x+5 + x-5 = 4(x^2 - 25)
2x = 4x^2 - 100

4x^2 - 2x - 100 = 0
x = (2 ± √1604)/8
x = 5.256 or x = a negative, which would make no sense

the speed of the boat in still water is appr 5.26 mph

a bout travels 1km upstream and 1 km downstream.it took the boat 1 hour for the round trip. the speed of the current is 2 kph. find the speed of the boat in still water.

To solve this problem, we can use the formula:

Speed of boat in still water = (Speed downstream + Speed upstream) / 2

Given that the speed of the stream is 5 km/hr, we can calculate the speed downstream and speed upstream.

Let's denote the speed of the boat in still water as "B" km/hr.

When the boat is going downstream (with the stream), its effective speed will be B + 5 km/hr.

When the boat is going upstream (against the stream), its effective speed will be B - 5 km/hr.

We know that the boat travels 1 km upstream and 1 km back, so the total distance traveled is 1 km + 1 km = 2 km.

The round trip time is given as 4 hours, so we can set up the following equation:

2 km / (B + 5 km/hr) + 2 km / (B - 5 km/hr) = 4 hours.

To simplify the equation, we can multiply throughout by (B + 5)(B - 5) to eliminate the denominators:

2(B - 5) + 2(B + 5) = 4(B^2 - 25).

Expanding and simplifying the equation, we get:

2B - 10 + 2B + 10 = 4B^2 - 100.

4B = 4B^2 - 100.

Rearranging the equation, we get:

4B^2 - 4B - 100 = 0.

Dividing through by 4, we get:

B^2 - B - 25 = 0.

To solve this quadratic equation, we can use the quadratic formula:

B = (-b ± √(b^2 - 4ac)) / (2a),

where a = 1, b = -1, and c = -25.

Plugging in the values, we have:

B = (-(-1) ± √((-1)^2 - 4(1)(-25))) / (2(1)),

B = (1 ± √(1 + 100)) / 2,

B = (1 ± √101) / 2.

The two possible solutions are:

B1 = (1 + √101) / 2,

B2 = (1 - √101) / 2.

However, since the speed of the boat cannot be negative, we can disregard the negative solution. Therefore, the speed of the boat in still water is (1 + √101) / 2 km/hr.

To find the speed of the boat in still water, we need to consider the relationship between the speed of the boat, the speed of the stream, and the time taken for the round trip.

Let's assume the speed of the boat in still water is "b" km/hr. Since the boat is traveling both upstream and downstream, the effective speed of the boat will be different in each direction due to the opposing or aiding stream.

When the boat is moving upstream, the speed of the stream subtracts from the speed of the boat, resulting in a slower effective speed. So the effective speed of the boat upstream can be calculated by subtracting the speed of the stream (5 km/hr) from the speed of the boat in still water, which is b - 5 km/hr.

Conversely, when the boat is moving downstream, the speed of the stream adds to the speed of the boat, resulting in a faster effective speed. So the effective speed of the boat downstream can be calculated by adding the speed of the stream (5 km/hr) to the speed of the boat in still water, which is b + 5 km/hr.

Now let's consider the time taken for the round trip. We know that the boat travels 1 km upstream and 1 km downstream. The time taken to travel upstream can be calculated by dividing the distance (1 km) by the effective speed of the boat upstream (b - 5 km/hr), which gives us 1 / (b - 5) hours.

Similarly, the time taken to travel downstream can be calculated by dividing the distance (1 km) by the effective speed of the boat downstream (b + 5 km/hr), which gives us 1 / (b + 5) hours.

According to the problem, the total time for the round trip is 4 hours. Therefore, the sum of the times taken to travel upstream and downstream should be equal to 4 hours:

1 / (b - 5) + 1 / (b + 5) = 4

Now we have a quadratic equation. We can solve this equation to find the speed of the boat in still water.

Unfortunately, quadratic equations cannot be solved easily using only algebraic methods. Therefore, it is recommended to use numerical methods or a graphing calculator to find the solution.