Answer :
Sure, let's work through this problem step-by-step to find the speed of the stream.
### Given Information:
- Speed of the boat in still water: \( 15 \) km/h
- Distance traveled upstream and downstream: \( 30 \) km each way
- Total time for the round trip: \( 30 \) minutes, which is \( 0.5 \) hours
### Let's denote:
- Speed of the stream as \( x \) km/h
- Speed of the boat upstream (against the current) as \( 15 - x \) km/h
- Speed of the boat downstream (with the current) as \( 15 + x \) km/h
### Time Calculation for Upstream and Downstream:
1. Time taken to travel upstream \( = \frac{\text{Distance}}{\text{Speed}} = \frac{30 \text{ km}}{15 - x \text{ km/h}} \)
2. Time taken to travel downstream \( = \frac{\text{Distance}}{\text{Speed}} = \frac{30 \text{ km}}{15 + x \text{ km/h}} \)
### Total Time for the Round Trip:
Given that the total time is 0.5 hours:
[tex]\[ \frac{30}{15 - x} + \frac{30}{15 + x} = 0.5 \][/tex]
### Solving for \( x \):
To solve this equation, we follow these steps:
1. Multiply through by the common denominator \((15 - x)(15 + x)\) to clear the denominators.
[tex]\[ 30 \cdot (15 + x) + 30 \cdot (15 - x) = 0.5 \cdot (15 - x)(15 + x) \][/tex]
2. Simplify the equation:
[tex]\[ 30(15 + x) + 30(15 - x) = 0.5 (225 - x^2) \][/tex]
[tex]\[ 450 + 30x + 450 - 30x = 0.5 (225 - x^2) \][/tex]
[tex]\[ 900 = 0.5 (225 - x^2) \][/tex]
3. Solve for \( x^2 \):
[tex]\[ 900 = 112.5 - 0.5x^2 \][/tex]
[tex]\[ 900 - 112.5 = -0.5x^2 \][/tex]
[tex]\[ 787.5 = -0.5x^2 \][/tex]
[tex]\[ x^2 = \frac{787.5}{-0.5} \][/tex]
[tex]\[ x^2 = -1575 \][/tex]
4. Taking the square root of both sides, we get:
[tex]\[ x = \sqrt{-1575} \][/tex]
This results in a complex number solution, indicating the possible values of \( x \):
[tex]\[ x = \pm 25.98i \][/tex]
### Conclusion:
The speed of the stream is a complex number, specifically [tex]\( \pm 25.98i \)[/tex] km/h. This indicates that under the given conditions, the stream speed in real numbers is not feasible, leading to an imaginary solution.
### Given Information:
- Speed of the boat in still water: \( 15 \) km/h
- Distance traveled upstream and downstream: \( 30 \) km each way
- Total time for the round trip: \( 30 \) minutes, which is \( 0.5 \) hours
### Let's denote:
- Speed of the stream as \( x \) km/h
- Speed of the boat upstream (against the current) as \( 15 - x \) km/h
- Speed of the boat downstream (with the current) as \( 15 + x \) km/h
### Time Calculation for Upstream and Downstream:
1. Time taken to travel upstream \( = \frac{\text{Distance}}{\text{Speed}} = \frac{30 \text{ km}}{15 - x \text{ km/h}} \)
2. Time taken to travel downstream \( = \frac{\text{Distance}}{\text{Speed}} = \frac{30 \text{ km}}{15 + x \text{ km/h}} \)
### Total Time for the Round Trip:
Given that the total time is 0.5 hours:
[tex]\[ \frac{30}{15 - x} + \frac{30}{15 + x} = 0.5 \][/tex]
### Solving for \( x \):
To solve this equation, we follow these steps:
1. Multiply through by the common denominator \((15 - x)(15 + x)\) to clear the denominators.
[tex]\[ 30 \cdot (15 + x) + 30 \cdot (15 - x) = 0.5 \cdot (15 - x)(15 + x) \][/tex]
2. Simplify the equation:
[tex]\[ 30(15 + x) + 30(15 - x) = 0.5 (225 - x^2) \][/tex]
[tex]\[ 450 + 30x + 450 - 30x = 0.5 (225 - x^2) \][/tex]
[tex]\[ 900 = 0.5 (225 - x^2) \][/tex]
3. Solve for \( x^2 \):
[tex]\[ 900 = 112.5 - 0.5x^2 \][/tex]
[tex]\[ 900 - 112.5 = -0.5x^2 \][/tex]
[tex]\[ 787.5 = -0.5x^2 \][/tex]
[tex]\[ x^2 = \frac{787.5}{-0.5} \][/tex]
[tex]\[ x^2 = -1575 \][/tex]
4. Taking the square root of both sides, we get:
[tex]\[ x = \sqrt{-1575} \][/tex]
This results in a complex number solution, indicating the possible values of \( x \):
[tex]\[ x = \pm 25.98i \][/tex]
### Conclusion:
The speed of the stream is a complex number, specifically [tex]\( \pm 25.98i \)[/tex] km/h. This indicates that under the given conditions, the stream speed in real numbers is not feasible, leading to an imaginary solution.
