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प्रश्न
A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse, then the tank will be filled in 30 minutes. How long would it take each pump to fill the tank by itself? (Use Cramer’s rule to solve the problem)
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उत्तर
Pump A fills `(1/x)^("th")` of the tank in 1 hour.
Pump B fills `(1/y)^("th")` of the tank in 1 hour.
Both can filled `(1/10)^("th")` of the tank in 1 hour.
`1/x + 1/y = 1/10` .........(1)
Pump B filled in 30 min.
`1/x - 1/y = 1/30` ..........(2)
Let a = `1/x`, b = `1/y`
a + b = `1/10`
a – b = `1/30`
Using Cramer’s rule
Δ = `|(1, 1),(1, -1)|` = 1 – 1 = – 2 ≠ 0
Δa = `|(1/10, 1),(1/30, -1)| = (-1)/10 - 1/30 = (-4)/30`
Δb = `|(1/10, 1),(1/30, -1)| = 1/13 - 1/10 = (-2)/30`
a = `Delta_"a"/Delta`
= `((-4)/30)/(-2)`
= `4/30 xx 1/2`
= `1/5`
⇒ x = 15
b = `Delta_"b"/Delta`
= `((-2)/30)/(-2)`
= `1/30`
⇒ y = 30
Pump A takes 15 minutes
Pump B takes 30 minutes
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