when only the cold water valve is opened, a washtub will fill in 8 minutes. When only the hot water valve is opened, the washtub will fill in 12 mintues. When the drain of the washtub is open, it will drain completely in 7 minutes. If both the hot and cold water valves are open and the drain is open, how long will it take for the washtub to fill?

Question

when only the cold water valve is opened, a washtub will fill in 8 minutes. When only the hot water valve is opened, the washtub will fill in 12 mintues. When the drain of the washtub is open, it will drain completely in 7 minutes. If both the hot and cold water valves are open and the drain is open, how long will it take for the washtub to fill?

Answer 1

cold 1/8 tub/min

hot 1/12 tub/min

drain -1/7 tub/min

so sum 1/8 +1/12 - 1/7 tub/min

= 3/24 + 2/24 - 1/7
= 5/24 - 1/7
= 35/168 - 24/168 = 11/168 tubs/min

= 168/11 min/tub = 15.3 min

Answer 2

To find out how long it takes for the washtub to fill when both the hot and cold water valves are open and the drain is open, we need to consider the rates at which each valve fills the washtub and the rate at which the drain empties the tub.

Let's calculate the rates first:

Rate of the cold water valve = 1/8 tubs per minute
Rate of the hot water valve = 1/12 tubs per minute
Rate of the drain = 1/7 tubs per minute

Now, when both the hot and cold water valves are open, their rates are added together:

Rate of both valves = 1/8 + 1/12 = 5/24 tubs per minute

Since the drain is also open, we need to subtract its rate from the combined rate:

Rate of filling with drain open = Rate of both valves - Rate of drain
= 5/24 - 1/7
= (35 - 24)/(24 * 7)
= 11/168 tubs per minute

Now, to find the time it takes to fill the washtub, we can use the formula:

Time = 1 / Rate

Time = 1 / (11/168)
Time = 168/11

Calculate this division to find the answer:

Time = 15.27 minutes (approximately)

Therefore, it will take approximately 15.27 minutes for the washtub to fill when both the hot and cold water valves are open and the drain is open.

Answer 3

To find the time it takes for the washtub to fill when both the hot and cold water valves are open and the drain is open, we need to determine the rate at which each valve fills the tub and the rate at which the drain empties it.

Let's assume that the washtub has a total capacity of 1 unit (You can assume any value, but it will cancel out in the calculations).

From the given information, we know:
- The cold water valve fills the tub in 8 minutes, so its filling rate is 1/8 units per minute.
- The hot water valve fills the tub in 12 minutes, so its filling rate is 1/12 units per minute.
- The drain empties the tub in 7 minutes, so its emptying rate is 1/7 units per minute.

When both the hot and cold water valves are open, their combined filling rate is the sum of their individual rates: 1/8 + 1/12 = 5/24 units per minute.

Since the drain is also open, the overall rate of filling the tub, taking into account the drain's emptying rate, is the difference between the filling rate and the emptying rate: 5/24 - 1/7 = (35 - 24) / (24 * 7) = 11 / (24 * 7) units per minute.

Now, to find the time it takes for the tub to fill completely, we can divide the total capacity of the tub (1 unit) by the overall filling rate:

Time = 1 / (11 / (24 * 7))
Time = 1 * (24 * 7) / 11
Time ≈ 12.545 minutes

Therefore, when both the hot and cold water valves are open and the drain is open, it will take approximately 12.545 minutes for the washtub to fill completely.