CBSE · Class 10 · Mathematics · Quadratic EquationsA 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.

Step-by-Step Solution:

1. Given Data:

• Speed of the motor boat in still water = $18 \text{ km/h}$
• Distance covered = $24 \text{ km}$

2. Assume Variables:

• Let the speed of the stream be $x \text{ km/h}$.

3. Determine Speeds:

• Speed of the boat upstream (धारा के प्रतिकूल चाल) = $(18 - x) \text{ km/h}$
• Speed of the boat downstream (धारा के अनुकूल चाल) = $(18 + x) \text{ km/h}$

4. Formulate the Equation:

• Time taken to go upstream ($T_1$) = $\frac{\text{Distance}}{\text{Upstream Speed}} = \frac{24}{18 - x}$
• Time taken to go downstream ($T_2$) = $\frac{\text{Distance}}{\text{Downstream Speed}} = \frac{24}{18 + x}$
• According to the question, the upstream time is 1 hour more than the downstream time: $$T_1 - T_2 = 1$$ $$\frac{24}{18 - x} - \frac{24}{18 + x} = 1$$

5. Solve the Quadratic Equation:

• Take out $24$ as a common factor and find the common denominator: $$24 \left[ \frac{1}{18 - x} - \frac{1}{18 + x} \right] = 1$$ $$24 \left[ \frac{(18 + x) - (18 - x)}{(18 - x)(18 + x)} \right] = 1$$ $$24 \left[ \frac{18 + x - 18 + x}{18^2 - x^2} \right] = 1$$ $$24 \left[ \frac{2x}{324 - x^2} \right] = 1$$ $$\frac{48x}{324 - x^2} = 1$$
• Cross-multiply to get: $$48x = 324 - x^2$$
• Rearranging into standard quadratic form ($ax^2 + bx + c = 0$): $$x^2 + 48x - 324 = 0$$

6. Factorization Method:

• We need two numbers whose product is $-324$ and sum is $48$.
• These numbers are $54$ and $-6$ (since $54 \times (-6) = -324$ and $54 + (-6) = 48$).
• Splitting the middle term: $$x^2 + 54x - 6x - 324 = 0$$ $$x(x + 54) - 6(x + 54) = 0$$ $$(x - 6)(x + 54) = 0$$

7. Find the Final Value:

• Either $x - 6 = 0 \implies x = 6$
• Or $x + 54 = 0 \implies x = -54$
• Since the speed of the stream cannot be negative, we reject $x = -54$.
• Therefore, $x = 6$.

Conclusion:

• The speed of the stream is $6 \text{ km/h}$.