Respuesta :
Answer:
The high power pump can fill the pool in 10 hours.
Step-by-step explanation:
Let the high power pump fill the pool in x hours, then the other pump will fill the pool in (x + 5) hours.
[tex]\text{Rate at which higher power pump fill the pool = }\frac{1}{x}\\\\\text{Rate at which other power pump fill the pool = }\frac{1}{(x+5)}[/tex]
Now, as given they together fill half of the the pool in 3 hours. So,
[tex]\frac{1}{x}+\frac{1}{(x+5)}=\frac{1}{3}\times \frac{1}{2}\text{.......here multiplication with }\frac{1}{2}\\\\\text{ indicates the time both pumps will take to fill the full pool.}\\\implies \frac{1}{x}+\frac{1}{(x+5)}=\frac{1}{6}\\\\\implies\frac{x+5+x}{x\cdot (x+5)}=\frac{1}{6}\\\\\implies x^2-7\cdot x-30=0\\\implies x=10,\thinspace or\thinspace x=-3[/tex]
But, time can not be negative. So, x = 10 hours
Therefore, The high power pump can fill the pool in 10 hours.
The high power pump take [tex]\boxed{4.4{\text{ hours}}}[/tex] to fill the pool.
Further explanation:
Explanation:
Let us assume that the time taken by the slower pump be [tex]y{\text{ hours}}.[/tex]
Therefore, the taken by the high speed pump is [tex]\left( {y - 5} \right){\text{ hours}}[/tex].
The water filled by slower pump in 1 hour can be expressed as,
[tex]{T_1} = \dfrac{1}{y}[/tex]
The water filled by high pump in 1 hour can be expressed as,
[tex]{T_2} = \dfrac{1}{y-5}[/tex]
The time taken by the two pumps if the fill together is 3 hours
[tex]\begin{aligned}\frac{1}{x} + \frac{1}{{x - 5}}&= \frac{1}{3}\\\frac{{x - 5 + x}}{{x\left( {x - 5} \right)}} &= \frac{1}{3}\\\frac{{2x - 5}}{{{x^2} - 5x}} &= \frac{1}{3}\\6x - 15 &= {x^2} - 5x\\0&= {x^2} - 11x + 15 \\\end{aligned}[/tex]
Solve the quadratic by discriminant formula equation.
[tex]\begin{aligned}x&= \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\\ x&= \frac{{ - \left( { - 11} \right) \pm \sqrt {{{\left( { - 11} \right)}^2} - 4\left( 1 \right)\left( {15} \right)} }}{{2 \times 1}} \\x &= \frac{{11 \pm \sqrt {61} }}{2}\\x &= \frac{{11 + 7.8}}{2}{\text{ or }}x = \frac{{11 - 7.8}}{2}\\x &= 9.4{\text{ or }}x = 1.6\\\end{aligned}[/tex]
The value of [tex]x = 1.6[/tex] cannot be considered.
Therefore, the time taken by the slower pump is [tex]\boxed{9.4{\text{ hours}}}[/tex].
The time taken by the high power pump is
[tex]\begin{aligned}{\text{Time}} &= x - 5\\&= 9.4 - 5\\&= 4.4\\\end{aligned}[/tex]
The high power pump take [tex]\boxed{4.4{\text{ hours}}}[/tex] to fill the pool.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Quadratic equation
Keywords: two water pumps, water pump, pump, quadratic equation, pool, 5 hours, high power pumps, working, slower pump.
