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The speed of a boat in still water is 11 km per hour. It can go 12 km upstream and return downstream to the original point in 2 hours and 45 minutes. Find the speed of stream.

Quadratic Equations

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Answer

Given,

Speed of boat = 11 km/hour

Let speed of stream be x km/hour.

Speed of boat (upstream) = (11 - x) km/hour

Speed of boat (downstream) = (11 + x) km/hour

Distance covered = 24 km (12 + 12)

Time taken = 2 hours 45 minutes = 2+45602 + \dfrac{45}{60} hours

By formula,

Time = DistanceSpeed\dfrac{\text{Distance}}{\text{Speed}}

Since, boat goes 12 km upstream and 12 km downstream.

∴1211−x+1211+x=2+4560⇒12(11+x)+12(11−x)(11−x)(11+x)=2+34⇒132+12x+132−12x121−x2=114⇒264121−x2=114⇒264×4=11(121−x2)⇒1056=1331−11x2⇒11x2=1331−1056⇒11x2=275⇒x2=27511⇒x2=25⇒x=25⇒x=±5.\therefore \dfrac{12}{11 - x} + \dfrac{12}{11 + x} = 2 + \dfrac{45}{60} \\[1em] \Rightarrow \dfrac{12(11 + x) + 12(11 - x)}{(11 - x)(11 + x)} = 2 + \dfrac{3}{4} \\[1em] \Rightarrow \dfrac{132 + 12x + 132 - 12x}{121 - x^2} = \dfrac{11}{4} \\[1em] \Rightarrow \dfrac{264}{121 - x^2} = \dfrac{11}{4} \\[1em] \Rightarrow 264 \times 4 = 11(121 - x^2) \\[1em] \Rightarrow 1056 = 1331 - 11x^2 \\[1em] \Rightarrow 11x^2 = 1331 - 1056 \\[1em] \Rightarrow 11x^2 = 275 \\[1em] \Rightarrow x^2 = \dfrac{275}{11} \\[1em] \Rightarrow x^2 = 25 \\[1em] \Rightarrow x = \sqrt{25} \\[1em] \Rightarrow x = \pm 5.

Since, speed cannot be negative.

∴ Speed of stream = 5 km/hour.

Hence, speed of stream = 5 km/hour.

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