A boat travels 4 km upstream and 4 km back. The time for the round trip is 4 hrs. The speed of the stream is 4 km/hr. What is the speed of the boat in still water?
since time = distance/speed,
4/(x-4) + 4/(x+4) = 4
x = 5.123
check:
4/1.123 + 4/9.123 = 3.56 + 0.44 = 4.00
speed of boat in still water ---- x km/h
so speed upstream = x-4 km/h
speed downstream = x+4 km/h
4/(x+4) + 4/(x-4) = 4 , lots of 4's there
multiply each term by (x+4)(x-4)
4(x-4) + 4(x+4) = 4(x+4)(x-4)
divide each term by 4 and expand
x-4 + x+4 = x^2 - 16
x^2 -2x -16 = 0
solve for x, use whatever method you like to solve quadratics
obviously rejecting the negative value of x
time = distance / rate
[4 / (s - 4)] + [4 / (s + 4)] = 4
(s + 4) + (s - 4) = (s + 4) (s - 4)
2 s = s^2 - 16
s^2 - 2s - 16 = 0
use the quadratic formula to find s
To find the speed of the boat in still water, we can use the formula:
Speed of boat in still water = (speed of boat downstream + speed of boat upstream) / 2
First, let's calculate the speed of the boat downstream and upstream.
Given:
Distance traveled upstream = 4 km
Distance traveled downstream = 4 km
Time taken for round trip = 4 hours
Speed of the stream = 4 km/hr
Let the speed of the boat in still water be 'B' km/hr.
1. Speed of boat downstream:
When the boat is traveling downstream, the speed of the stream adds to the speed of the boat, so the effective speed is:
B + 4 km/hr
2. Speed of boat upstream:
When the boat is traveling upstream, the speed of the stream subtracts from the speed of the boat, so the effective speed is:
B - 4 km/hr
Now we can calculate the time taken for each leg of the journey.
Time taken to travel upstream = Distance / Speed
Time taken to travel upstream = 4 km / (B - 4) km/hr
Time taken to travel downstream = Distance / Speed
Time taken to travel downstream = 4 km / (B + 4) km/hr
Since the total round trip time is given as 4 hours, the sum of the upstream and downstream times should be equal to 4 hours:
4 km / (B - 4) km/hr + 4 km / (B + 4) km/hr = 4 hours
To solve this equation, we can multiply through by (B - 4)(B + 4) to remove the denominators:
4 km(B + 4) + 4 km(B - 4) = 4 hours(B - 4)(B + 4)
Simplifying the equation further:
4B + 16 + 4B - 16 = 4(B^2 - 16)
8B = 4B^2 - 64
4B^2 - 8B - 64 = 0
Now, we can solve this quadratic equation to find the value of B, which will be the speed of the boat in still water. We can use the quadratic formula:
B = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 4, b = -8, and c = -64.
B = (-(-8) ± √((-8)^2 - 4 * 4 * -64)) / (2 * 4)
B = (8 ± √(64 + 1024)) / 8
B = (8 ± √1088) / 8
B = (8 ± √(16 * 68)) / 8
B = (8 ± 4√17) / 8
Now, we simplify the expression further:
B = (2 ± √17) / 2
Hence, the two possible solutions for the speed of the boat in still water are:
1. B = (2 + √17) / 2
2. B = (2 - √17) / 2
Therefore, the speed of the boat in still water can be (2 + √17) / 2 km/hr or (2 - √17) / 2 km/hr, depending on the context of the problem.