Question

A motorboat whose speed in still water is 15km/hour, goes 30km downstream and
returns back to starting point in a total time of 4 hours and 30 minutes. Find the speed
of the stream. ​

Answer 1

Speed of motorboat in still water is 15 km/h.

Distance travelled by motorboat is 30 km.

Time taken to returns back at starting point is 4 hr 30 min.

We have to find, speed of stream.

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☯ Let speed of stream be x km/h.

Therefore,

Speed of motorboat downstream = (15 + x) km/h.

Speed of motorboat upstream = (15 - x) km/h.

$\underline{\bigstar\:\boldsymbol{According\:to\: Question\::}}$

$:\implies\sf \dfrac{30}{(15 + x)} + \dfrac{30}{(15 - x)} = 4\dfrac{1}{2}$

$:\implies\sf \dfrac{30}{(15 + x)} + \dfrac{30}{(15 - x)} = \dfrac{9}{2}$

$:\implies\sf \dfrac{30(15 - x) + 30(15 + x)}{(15 + x)(15 - x)} = \dfrac{9}{2}$

$:\implies\sf \dfrac{30(15 - x) + 30(15 + x)}{225 - x^2} = \dfrac{9}{2}\qquad\bigg\lgroup\bf (a + b)(a - b) = a^2 - b^2 \bigg\rgroup$

$:\implies\sf \dfrac{30(15 - x + 15 + x)}{225 - x^2} = \dfrac{9}{2}$

$:\implies\sf \dfrac{30(15\; \cancel{ - x} + 15 \;\cancel{+ x})}{225 - x^2} = \dfrac{9}{2}$

$:\implies\sf \dfrac{30(15 + 15)}{225 - x^2} = \dfrac{9}{2}$

$:\implies\sf \dfrac{30(30)}{225 - x^2} = \dfrac{9}{2}$

$:\implies\sf \dfrac{900}{225 - x^2} = \dfrac{9}{2}$

$:\implies\sf 2(900) = 9(225 - x^2)$

$:\implies\sf 1800 = 9(225 - x^2)$

$:\implies\sf \cancel{ \dfrac{1800}{9}} = 225 - x^2$

$:\implies\sf 200 = 225 - x^2$

$:\implies\sf x^2 = 225 - 200$

$:\implies\sf x^2 = 25$

$:\implies\sf \sqrt{x^2} = \sqrt{25}$

$:\implies \bold{\underline{\boxed{\sf{\purple{x = 5}}}}}\;\bigstar$

$\therefore\;\sf \underline{Speed\;of\; stream\;is\; \bold{5\;km/h}.}$