a motorboat whose speed is 18

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Question 8 - Examples - Chapter 4 Class 10 Quadratic Equations

Last updated at April 16, 2024 by Teachoo

Question 8 A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. Given that speed of the boat = 18 km/ hr. Let the speed of the stream = x km / hr. Given that Time taken upstream is 1 hour more than time taken downstream Time upstream = Time downstream + 1 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 24((18 + 𝑥) − (18 − 𝑥))/((18 − 𝑥)(18 + 𝑥)) = 1 24(18 + 𝑥 − 18 + 𝑥)/((18 − 𝑥)(18 + 𝑥)) = 1 24(2𝑥)/((18 − 𝑥)(18 + 𝑥)) = 1 48𝑥/((18 − 𝑥)(18 + 𝑥)) = 1 48x = (18 – x) (18 + x) 48x = 182 – x2 48x = 324 – x2 x2 + 48x – 324 = 0 Comparing equation with ax2 + bx + c = 0, Here a = 1, b = 48, c = –324 We know that D = b2 – 4ac D = (48)2 – 4 × 1 × (–324) D = 2304 + 4 × 324 D = 2304 + 1296 D = 3600 So, the roots to equation are x = (−𝑏 ± √𝐷)/2𝑎 Putting values x = (−(48) ± √3600)/(2 × 1) x = (− 48 ± √(60 × 60))/(2 × 1) x = (− 48 ± 60)/2 Solving So, x = 6 & x = – 54 Since, x is the speed , so it cannot be negative So, x = 6 is the solution of the equation Therefore, speed of the stream (x) = 6 km /hr.

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

The question is a real-life application of linear equations in two variables .

Answer: The speed of the stream is 6 km/hr.

Let's explore the water currents.

Explanation:

Let the speed of the stream be x km/hr

Given that, the speed boat in still water is 18 km/hr.

Sspeed of the boat in upstream = (18 - x) km/hr

Speed of the boat in downstream = (18 + x) km/hr

It is mentioned that the boat takes 1 hour more to go 24 km upstream than to return downstream to the same spot

Therefore, One-way Distance traveled by boat (d) = 24 km 

Hence, Time in hour 

T upstream  = T downstream   + 1

[distance / upstream speed ] = [distance / downstream speed]     + 1

[ 24/ (18 - x) ] = [ 24/ (18 + x) ] + 1 

[ 24/ (18 - x) - 24/ (18 + x) ] = 1 

24 [1/ (18 - x) - 1/(18 + x) ] = 1

24 [ {18 + x - (18 - x) } / {324 - x 2 } ] = 1

24 [ {18 + x - 18 + x) } / {324 - x 2 } ] = 1

⇒ 24 [ {2}x / {324 - x 2 } ] = 1

⇒ 48x = 324 - x 2

⇒ x 2  + 48x - 324 = 0

⇒ x 2  + 54x - 6x - 324 = 0   ----------> (by splitting the middle-term)

⇒ x(x + 54) - 6(x + 54) = 0

⇒ (x + 54)(x - 6) = 0

⇒ x = -54  or 6

As speed to stream can never be negative, we consider the speed of the stream (x) as 6 km/hr.

Thus, the speed of the stream is 6 km/hr.

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A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

• Mathematics
• Solution of a Quadratic Equation by...

A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Given: speed of boat in still water = 18 k m / h r let speed of the stream = s speed of boat upstream = speed of boat in still water - speed of stream = 18 − s speed of boat down stream = speed of boat in still water + speed of stream = 18 + s time taken for upstream = time taken to cover downstream + 1 ⇒ d i s t a n c e u p s t r e a m s p e e d u p s t r e a m = d i s t a n c e d o w n s t r e a m s p e e d d o w n s t r e a m + 1 ⇒ 24 18 − s = 24 18 + s + 1 ⇒ 24 ( 18 + s ) = 24 ( 18 − s ) + ( 18 − s ) ( 18 + s ) ⇒ s 2 + 48 s − 324 = 0 ⇒ s 2 + 54 s − 6 s − 324 = 0 ⇒ ( s + 54 ) ( s − 6 ) = 0 ⇒ s = 6 , − 54 ⇒ s ≠ − 54 thus, s = 6 k m / h r , speed of steam cannot be negative..

A motor boat whose speed is 18 k m h r in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream in km/hr.​

A motor boat has a speed of 18 k m h r in still water. It takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream in k m / h r .

7 k m / h r

6 k m / h r

4 k m / h r

5 k m / h r

The correct option is B 6 k m / h r Let the speed of the stream be ' x ' and the time required to travel upstream be ' t '. T i m e = D i s t a n c e S p e e d Speed of boat in upstream = Speed of boat - Speed of stream = 18 - x and, speed of boat in downstream = speed of boat + speed of stream = 18 + x ∴ From the given data, t = 24 ( 18 + x ) ---------------(1) t + 1 = 24 ( 18 − x ) ------------(2) Subtracting (1) from (2), we get 1 = 24 [ 1 18 − x − 1 18 + x ] ⇒ 1 = 24 [ 2 x ( 18 2 − x 2 ) ] ⇒ 18 2 − x 2 = 48 x ⇒ x 2 + 48 x − 18 2 = 0 Now, the equation x 2 + 48 x − 18 2 = 0 can also be written as: x 2 + 2 ( 24 ) x + 24 2 − 24 2 − 18 2 = 0 ⇒ ( x + 24 ) 2 − 900 = 0 ⇒ ( x + 24 ) 2 − 30 2 = 0 ⇒ ( x + 24 ) 2 = 30 2 ⇒ x + 24 = ± 30 (Taking square root on both sides) ⇒ x = − 54 or x = 6 The speed of the stream cannot be negative. So, the speed of the stream = 6 k m / h r

A motor boat has a speed of 18 k m h r in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream in k m / h r .

A motor boat whose speed is 18 k m h r in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream in km/hr.​