Question

A Swimming pool is filled with three pipes with three pipes with uniform flow. The first two pipes operating simultaneously,fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool seperately.

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Answer 1

Answer:

The time taken by the first pipe to fill the pool = 15 hours.

The time taken by the second pipe to fill the pool = (15−5)

hours = 10 hours.

The time taken by the third pipe to fill the pool = (15−9)

hours = 6 hours.

So, the correct answer is “Option B”.

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

• A Swimming pool is filled with three pipes with three pipes with uniform flow.

• The first two pipes operating simultaneously,fill the pool in the same time during which the pool is filled by the third pipe alone.

• The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe.

Let assume that

• V be the volume of the pool.

• Number of hours required by second pipe alone to fill the pool be x hours.

So,

• Time taken by first pipe alone to fill the pool be x + 5 hours.

• Time taken by third pipe alone to fill the pool be x - 4 hours.

Further assume that

• The time taken by first and second pipe to fill the pool be 't' hours.

So,

• Time taken by third pipe to fill the pool alone be t hours.

So,

$\rm :\longmapsto\:\bigg[\dfrac{V}{x + 5} + \dfrac{V}{x}\bigg]t = \dfrac{V}{x - 4} t$

$\rm :\longmapsto\:\dfrac{1}{x + 5} + \dfrac{1}{x} = \dfrac{1}{x - 4}$

$\rm :\longmapsto\:\dfrac{x + x + 5}{(x + 5)x} = \dfrac{1}{x - 4}$

$\rm :\longmapsto\:\dfrac{2x + 5}{(x + 5)x} = \dfrac{1}{x - 4}$

$\rm :\longmapsto\:(2x + 5)(x - 4) = x(x + 5)$

$\rm :\longmapsto\: {2x}^{2} + 5x - 8x - 20 = {x}^{2} + 5x$

$\rm :\longmapsto\: {2x}^{2} + 5x - 8x - 20 - {x}^{2} - 5x = 0$

$\rm :\longmapsto\: {x}^{2}- 8x - 20 = 0$

$\rm :\longmapsto\: {x}^{2}- 10x + 2x- 20 = 0$

$\rm :\longmapsto\:x(x - 10) + 2(x - 10) = 0$

$\rm :\longmapsto\:(x - 10)(x+ 2) = 0$

$\bf\implies \:x = 10 \: \: or \: \: x = - 2 \: \: \{rejected \}$

So,

• Time taken by first pipe alone to fill the pool be 15 hours.

• Time taken by second pipe alone to fill the pool be 10 hours.

• Time taken by third pipe alone to fill the pool be 6 hours.

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Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac