To solve this problem, we need to break it down step by step. We know that the speed of the motorboat in still water is 24 km/hr. Let’s denote the speed of the stream as ‘x’ km/hr. Therefore, the effective speed of the motorboat going upstream (against the current) is (24 – x) km/hr, and going downstream (with the current) is (24 + x) km/hr.
The time taken to travel a distance is calculated using the formula: Time = Distance / Speed.
1. **Upstream:** The time taken to go 32 km upstream is: Tup = 32 / (24 – x)
2. **Downstream:** The time taken to go 32 km downstream is: Tdown = 32 / (24 + x)
According to the problem, the upstream journey takes 1 hour more than the downstream journey. This gives us the equation:
Tup = Tdown + 1
Substituting the expressions for time into this equation:
32 / (24 – x) = 32 / (24 + x) + 1
Now, we will solve this equation for ‘x’. First, eliminate the fractions by multiplying through by (24 – x)(24 + x):
32(24 + x) = 32(24 – x) + (24 – x)(24 + x)
This simplifies to:
32x + 768 = -32x + 768 + 576 – x^2
Combining like terms gives us:
64x + x^2 = 576
Rearranging the equation leads to:
x^2 + 64x – 576 = 0
Now, we can use the quadratic formula x = (-b ± √(b² – 4ac)) / (2a), where a = 1, b = 64, and c = -576.
Calculating the discriminant:
Δ = 64² – 4(1)(-576) = 4096 + 2304 = 6400
Taking the square root gives us: √6400 = 80
Using the quadratic formula:
x = (-64 ± 80) / 2
This results in two potential solutions:
x = 8 (valid speed of the stream) and x = -72 (not valid since speed cannot be negative).
Thus, the speed of the stream is 8 km/hr.