Let speed of stream =x

speed of motorboat = 24 km/h

distance= 32km

time taken by motorboat to go downstream= 32/(x+ 24) h

time taken to go upstream = 32/(24-x)

32/(x+24) + 1 = 32/(24- x)

=32/(24- x) - 32/(x+24) = 1

=32(x+24) - 32(24-x) = (24-x)(x+24)

=32x + 32*24 - 32*24 + 32x = 242 - x2

=64x= 576 - x2

= x2 +64x - 576=0

=x2 +72x - 8x -576 = 0

= x(x+72) - 8(x-72) =0

= x -8 = 0 or x+72=0

= x=8km/h as x cant be -72 which is negative

so speed of stream is 8km/h.

Problem: Find the speed of the stream when a motorboat takes 1 hour more to travel 32 km upstream than to return downstream to the same spot, given that the speed of the motorboat in still water is 24 km/h.

Understanding the problem:
To solve the problem, we need to find the speed of the stream. We know that the speed of the boat in still water is 24 km/h. We also know that the boat takes 1 hour more to travel 32 km upstream than to return downstream to the same spot.

Assumptions:
The problem assumes that the speed of the boat is constant and there are no external factors affecting its speed.

Solution:

Let the speed of the stream be x km/h.

Step 1: Calculate the speed of the boat upstream and downstream
- Let the speed of the boat upstream be y km/h.
- The speed of the boat downstream is the sum of the speed of the boat in still water and the speed of the stream, i.e., 24 + x km/h.
- The speed of the boat upstream is the difference between the speed of the boat in still water and the speed of the stream, i.e., 24 - x km/h.

Step 2: Calculate the time taken to travel upstream and downstream
- Let the time taken to travel downstream be t hours.
- The time taken to travel upstream is t + 1 hours.
- Using the formula, distance = speed x time, we get:
- Distance downstream = (24 + x) t km
- Distance upstream = (24 - x) (t + 1) km
- Distance downstream = Distance upstream + 32 km (as the boat returns to the same spot)

Step 3: Solve for x
- Substituting the values obtained in step 2, we get:
- (24 + x) t = (24 - x) (t + 1) + 32
- Simplifying the equation, we get:
- 2xt + x = 32
- x(2t + 1) = 32
- Dividing both sides by (2t + 1), we get:
- x = 32/(2t + 1) km/h

Step 4: Find the value of t
- Substituting the value of x in the equation obtained in step 2, we get:
- 24t + 12x = 32
- Substituting the value of x in the equation obtained in step 3, we get:
- 24t + 12(32/(2t + 1)) = 32
- Simplifying the equation, we get:
- 24t(2t + 1) + 384 = 32(2t + 1)
- 48t^2 + 24t + 384 = 64t + 32
- 48t^2 - 40t - 352 = 0
- 12t^2 - 10t - 88 = 0
- Solving the quadratic equation, we get:
- t = 2.75 or t