Question

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)

Answer 1

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

Δ_a = `|(1/10, 1),(1/30, -1)| = (-1)/10 - 1/30 = (-4)/30`

Δ_b = `|(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