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Answers
EXPLANATION.
Question = 20.
A motor boat whose speed = 18 km/hr. in still water.
Takes 1 hours more to go 24 km upstream.
Return downstream to the same spot.
As we know that,
For upstream,
Speed = (18 - x) km/hr.
Distance of upstream = 24 km.
As we know that,
Formula of :
Distance = Speed x Time.
Time = Distance/Speed.
⇒ T₁ = 24/(18 - x). - - - - - (1).
For Downstream.
Speed = (18 + x)km/hr.
Distance = 24km.
As we know that,
Formula of :
Distance = Speed x Time.
Time = Distance/Speed.
⇒ T₂ = 24/(18 + x). - - - - - (2).
According to the question,
⇒ T₁ = T₂ + 1.
Put the value in the equation, we get.
⇒ 24/(18 - x) = 24/(18 + x) + 1.
⇒ 24/(18 - x) - 24/(18 + x) = 1.
⇒ 24(18 + x) - 24(18 - x)/(18 - x)(18 + x) = 1.
⇒ 432 + 24x - (432 - 24x)/(324 - x²) = 1.
⇒ 432 + 24x - 432 + 24x/(324 - x²) = 1.
⇒ 24x + 24x/(324 - x²) = 1.
⇒ 48x = 324 - x².
⇒ x² + 48x - 324 = 0.
Factorizes the equation into middle term splits, we get.
⇒ x² + 54x - 6x - 324 = 0.
⇒ x(x + 54) - 6(x + 54) = 0.
⇒ (x - 6)(x + 54) = 0.
⇒ x = 6 and x = - 54.
⇒ x ≠ - 54 [speed ≠ negative].
⇒ x = 6.
Speed of stream = 6km/hr.
Question = 21.
Train travel at a certain speed of 63km.
Then travel a distance of 72km.
At an average speed of 6 km/hr. more than the original speed.
If it takes 3 hours to complete total journey.
As we know that,
Let the original average speed of train = x km/hr.
Time taken to cover speed of 63 km = 63/x. hours
Time taken to cover speed of 72 km = 72/x. hours
According to the question,
⇒ 72/(x + 6) hours.
⇒ 63/x + 72/(x + 6) = 3.
⇒ 21/x + 24/(x + 6) = 1.
⇒ 21(x + 6) + 24(x)/(x)(x + 6) = 1.
⇒ 21x + 126 + 24x/(x² + 6x) = 1.
⇒ 45x + 126/(x² + 6x) = 1.
⇒ 45x + 126 = x² + 6x.
⇒ x² + 6x - 45x - 126 = 0.
⇒ x² - 39x - 126 = 0.
Factorizes the equation into middle term splits, we get.
⇒ x² - 42x + 3x - 126 = 0.
⇒ x(x - 42) + 3(x - 42) = 0.
⇒ (x + 3)(x - 42) = 0.
⇒ x = - 3 and x = 42.
⇒ x ≠ - 3 [ speed ≠ negative].
⇒ x = 42 km/hr.
The original average speed = 42km/hr.