Let the speed of river be x km/hr down stream.
Time = Distance / speed
Since river flows downstream, speed of boat down stream = Speed of boat + speed of river = (15 + x).
Since river flows downstream, speed of boat upstream = Speed of boat - speed of river = (15 - x).
Time Upstream - Time Downstream = 75 minutes
Time Upstream = 45 / (15 - x)
Time Downstream = 45 / (15 + x)
75 minutes = 75/60 = 5/4 hours
Time Upstream - Time Downstream = 75 minutes = 5/4 hours
45 / (15 - x) - 45 / (15 + x) = 5/4 Divide both sides by 45
1 / (15 - x) - 1 / (15 + x) = (5/4)*(1/45)
1 / (15 - x) - 1 / (15 + x) = 1/36
((15 + x) - (15 -x)) / (15-x)(15+x) = 1/36
(15 +x - 15 +x) / (15-x)(15+x) = 1/36
2x / (15-x)(15+x) = 1/36
(15-x)(15+x) = 2x*36
(15-x)(15+x) = 72x
225 - x² = 72x
0 = x² + 72x -225
x² + 72x -225 = 0 This is a
quadratic function, use a calculator that can solve the function, by
inputting the function.
x = 3, or -75. Since we are solving for speed, we can not have negative values.
x = 3 is the only valid solution.
Speed of the river = 3 km/hr downstream.
Copyright.
This advice is based upon your knowing the first ten or so perfect squares: {1, 4, 9, 16, ... } and their square roots. For example, the sqrt of 16 is 4.
I'd take the given number and determine where it stands among this list of perfect squares. For example, 20 would be between perfect squares 16 and 25.
We could surmise that the sqrt of 20 would be betwen the square roots of 16 and 25, which are, of course, 4 and 5.
We could do a bit better at estimating the sqrt of that number by interpolation. Note that sqrt(20) is closer to 4 than to 5. We could then surmise that the sqrt of 20 is a bit closer to 4 than to 5, e. g., sqrt(20) is approximately 4.4.
Using a calculator as a check: sqrt(20)= 4.47. Thus, our estimate was a bit on the low side: 4.4 instead of 4.47.
