Question

40 points..

Two water taps can fill a tank together in 75/8 hours. The tap with the smaller diameter takes 10 hours more than the tap with the larger diameter to fill the tank seperately. Find the time taken by each tap to fill the tank seperately.

Answer 1

Let the time taken by the tap with smaller diameter be T1 and by the tap with larger diameter be T2.

The, T1 - T2 = 10

T2 = T1 - 10 ---> ( i )

In 1hr, water filled by the tap with smaller diameter = 1 / T1

In 1hr, part of water filled by the tap with larger diameter = 1 / T2 = 1 / ( T1 - 10 )

Time taken by both the taps to fill the tank together = 75 / 8 hrs.

But, in 1hr the tank will fill the part of water by 1 / 75 / 8 = 8 / 75

1 / T1 + ( 1 / ( T1 - 10 ) = 8 / 75

( T1 - 10 ) + T1 / T1 ( T1 - 10 ) = 8 /75

2T1 - 10 / ( T1^2 - 10 T1 ) = 8 /75

2 ( T1 - 5 ) / ( T1 ^2 - 10 T1) = 8 / 75

( T 1 - 5 ) / ( T1^2 - 10 T1 ) = 4 / 75

75 T1 - 375 = 4T1 ^2 - 40 T1

0 = 4 T1^2 - 40 T1 - 75 T1 + 375

0 = 4 T1^2 - 115 T1 + 375

0 = 4 T1 ^2 - 100 T1 - 15 T1 + 375

0 = 4 T1 ( T1 - 25 ) - 15 ( T1 - 25 )

0 = ( 4 T1 - 15 ) ( T1 - 25 )

( 4 T1 - 15 ) = 0

T1 = 15 / 4

( T1 - 25) = 0

T1 = 25

So, time taken by the tap with smaller diameter is 25 hrs.

Time taken by the tap with larger diameter = ( T1 - 10 ) = 25 - 10 = 15 hrs.

Answer 2

$\bf\:Answer:$

15 hours, 25 hours

$\bf\:Step-by-step -explanation:$

Let the time taken by the larger pipe to fill the tank be 'x' hours.

Then,Time taken by the smaller pipe = (x + 10) hr.

∴ Part of tank filled by larger pipe = (1/x)

∴ Part of tank filled by smaller pipe = (1/x + 10).

Given that taps can fill a tank together in 75/8 hours.

⇒ (1/x) + (1/x + 10) = 8/75

⇒ (x + 10 + x)/x(x + 10) = 8/75

⇒ (2x + 10)/x(x + 10) = 8/75

⇒ 75(2x + 10) = 8x² + 80x

⇒ 150x + 750 = 8x² + 80x

⇒ 8x² - 70x - 750 = 0

⇒ 4x² - 35x - 375 = 0

⇒ 4x² - 60x + 25x - 375 = 0

⇒ 4x(x - 15) + 25(x - 15) = 0

⇒ (4x + 25)(x - 15) = 0

⇒ x = -25/4, 15.

x ≠ -25/4 as time cannot be negative.

∴ x = 15.

∴ x + 10 = 25 hours.

Therefore:

$<b>$Time taken by larger tap = 15 hours.

Time taken by smaller tap = 25 hours.

Hope it helps!