A river flows at the rate of 7km/hr. A patrol boat travels 46 km upriver and returns in a total time of 15hr. What is the speed of the boat in still water?
time = distance/speed
so, if the boat's speed is s, we have
46/(s+7) + 46/(s-7) = 15
Now just solve for s
To find the speed of the boat in still water, you need to use the concept of relative speed and the formula for time, distance, and speed.
Let's assume the speed of the boat in still water is "x" km/hr, and the speed of the river flow is 7 km/hr.
When the boat travels upstream (against the flow of the river), the effective speed of the boat will be reduced by the speed of the river flow. So the relative speed between the boat and the river is (x - 7) km/hr.
When the boat travels downstream (with the flow of the river), the effective speed of the boat will be increased by the speed of the river flow. So the relative speed between the boat and the river is (x + 7) km/hr.
Now, let's calculate the time taken for the upstream and downstream trips:
Time taken for the upstream trip: Distance / Relative speed
Time taken for the downstream trip: Distance / Relative speed
Given that the distance for each trip is 46 km and the total time for both trips is 15 hours, we can set up the following equation:
46 / (x - 7) + 46 / (x + 7) = 15
To solve this equation, we can simplify it by finding a common denominator:
(46(x + 7) + 46(x - 7)) / ((x - 7)(x + 7)) = 15
Now, simplify the equation further:
(46x + 322 + 46x - 322) / (x^2 - 49) = 15
92x / (x^2 - 49) = 15
Cross-multiply:
92x = 15x^2 - 735
Rearrange the equation:
15x^2 - 92x - 735 = 0
Now, you can solve this quadratic equation to find the value(s) of x (the speed of the boat in still water).