a boat travels 12 km upstream and back in 1 hour and 45 min.if the speed of the current is 3 km/h throughout,find the speed of the boat in still water,giving your answer correct to 3 decimal place.
NOTE:(Answer is 14.3 km/h,so can any one show me the solution).
The speed in still water is X km/h, so the speed upstream is (X-3) km/h (slower than X because it's upstream), and the speed downstream is (X+3) km/h (faster than X because it's downstream).
Travelling at (X-3) km/h for 12 km will take 12/(X-3) hours, and travelling back again at (X+3) km/h for 12 km will take another 12/(X+3) hours.
But the whole journey takes 1.75 hours, so 12/(X-3) + 12/(X+3) = 1.75 = 7/4. We need to solve this for X. After a bit of algebraic shuffling of the above terms, we end up with 7X²-96X-63=0. One of the two solutions of this is negative and therefore impossible; the other one should be the answer you're looking for.
To find the speed of the boat in still water, we can use the concept of relative velocities.
Let's assume the speed of the boat in still water is B km/h.
When the boat is traveling upstream, the effective speed is reduced by the speed of the current. So, the speed of the boat relative to the ground is (B - 3) km/h.
Similarly, when the boat is traveling downstream, the effective speed is increased by the speed of the current. So, the speed of the boat relative to the ground is (B + 3) km/h.
According to the given information, the boat travels 12 km upstream and back in a total of 1 hour and 45 minutes, which is equal to 1.75 hours.
Let's calculate the time taken by the boat to travel 12 km upstream:
Time taken = Distance / Speed
Time taken upstream = 12 / (B - 3)
Similarly, the time taken by the boat to travel 12 km downstream is:
Time taken downstream = 12 / (B + 3)
The total time taken for the entire round trip is 1.75 hours:
Time taken upstream + Time taken downstream = 1.75
Substituting the previously derived equations, we get:
12 / (B - 3) + 12 / (B + 3) = 1.75
To solve this equation, we can multiply both sides by (B - 3) * (B + 3) to eliminate the denominators:
12(B + 3) + 12(B - 3) = 1.75(B - 3)(B + 3)
Simplifying the equation:
12B + 36 + 12B - 36 = 1.75(B^2 - 9)
24B = 1.75B^2 - 15.75
1.75B^2 - 24B - 15.75 = 0
To solve this quadratic equation, we can use the quadratic formula:
B = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1.75, b = -24, and c = -15.75.
Substituting the values, we get:
B = (-(-24) ± √((-24)^2 - 4 * 1.75 * -15.75)) / (2 * 1.75)
Simplifying further:
B = (24 ± √(576 + 138.375)) / 3.5
B = (24 ± √(714.375)) / 3.5
B = (24 ± 26.740) / 3.5
Solving both positive and negative cases:
For B = (24 + 26.740) / 3.5 = 50.740 / 3.5 = 14.496 km/h (rounded to three decimal places)
For B = (24 - 26.740) / 3.5 = -2.740 / 3.5 = -0.782 km/h (rounded to three decimal places)
Since speed cannot be negative, the speed of the boat in still water is 14.496 km/h. Rounded to three decimal places, it is approximately 14.3 km/h.