A water reservoir has two inlets and one outlet. Through the inlet it can be filled in 3 hours and 3 hours 45 minutes respectively. It can be emptied completely in 1 hour by the outlet. If the two inlets are opened at 01:00 pm and 02:00 pm respectively and the outlet at 03:00 pm then it will be emptied at.
(a) 05 : 55 pm (b) 05 : 00 pm (c) 05 : 20 pm (d) 05 : 30 pm (e) none of these
To solve this problem, we need to find the net inflow rate of water into the reservoir per hour.
The first inlet fills the reservoir in 3 hours, so its inflow rate is 1/3 per hour.
The second inlet fills the reservoir in 3 hours 45 minutes, which can be written as 3.75 hours. So its inflow rate is 1/3.75 per hour.
The outlet empties the reservoir in 1 hour, so its outflow rate is 1 per hour.
Now, let's calculate the net inflow rate:
Inflow rate = (Inlet 1 rate) + (Inlet 2 rate)
= 1/3 + 1/3.75
= 0.333 + 0.267
= 0.60 per hour
This means that the net inflow rate of water into the reservoir is 0.60 units per hour.
Next, let's determine the time it takes for the reservoir to be emptied.
From 01:00 pm to 02:00 pm, the first inlet is open for 1 hour, so the water level in the reservoir increases by 0.60 units.
From 02:00 pm to 03:00 pm, the second inlet is open for 1 hour, so the water level in the reservoir increases by another 0.60 units.
From 03:00 pm onwards, the outlet is open, and the water level decreases at a rate of 1 unit per hour.
Since the water level increased by 1.20 units in the first two hours and then decreased by 1 unit per hour, it will take an additional 1.20 hours to empty the reservoir completely.
Thus, the reservoir will be emptied at 04:20 pm.
Therefore, the correct answer is (c) 05 : 20 pm.
To solve this problem, we need to determine the net rate at which the reservoir is being filled or emptied at different time intervals.
Let's calculate the filling rate of each inlet:
- The first inlet fills the reservoir in 3 hours, so its filling rate is 1/3 of the reservoir's capacity per hour.
- The second inlet fills the reservoir in 3 hours 45 minutes, which is equivalent to 3.75 hours. Therefore, its filling rate is 1/3.75 of the reservoir's capacity per hour.
Now let's calculate the emptying rate of the outlet:
- The outlet empties the reservoir in 1 hour, so its emptying rate is 1/1 of the reservoir's capacity per hour.
To find the net filling or emptying rate, we need to subtract the emptying rate from the sum of the filling rates of the two inlets:
Net filling rate = (1/3 + 1/3.75) - 1/1 = (4/15 + 4/15) - 1 = 8/15 - 1 = -7/15
Since the net filling rate is negative, it means that the reservoir is being emptied.
Now, let's calculate the time it takes to empty the reservoir using this net filling rate. From 2:00 pm to 3:00 pm, the time interval is 1 hour.
Let's assume that the reservoir's capacity is 15 units (this assumption will not affect the final answer).
During the 1 hour interval, the net emptying rate is -7/15 units per hour.
Therefore, the amount of water emptied from the reservoir during this 1-hour interval is (7/15) * 15 = 7 units.
Since the reservoir is being emptied, the level of water left in the reservoir will be 15 - 7 = 8 units.
So at 3:00 pm, there are still 8 units of water in the reservoir.
Now let's find out the time it takes for the remaining 8 units of water to be emptied.
The emptying rate is 1 unit per hour. So it will take 8 hours to empty the remaining 8 units of water.
Therefore, the reservoir will be completely emptied at 3:00 pm + 8 hours = 11:00 pm.
Since none of the options provided is 11:00 pm, the correct answer is (e) none of these.
let the volume of the reservoir be w units
let the total time from noon to empty time be t:00 hrs
rate of first inlet = w/3
rate of 2nd inlet = w/3.75
rate of outlet = w/1
time outlet is open = t-3
time 2nd inlet open = t-2
time 1st inlet is open = t -1
(w/3)(t-1) + (w/3.75)(t-2) - w(t-3) = 0 <--- it is empty
divide by w
(t-1)/3 + (t-2)/3.75 - t + 3 = 0
multiply by 11.25
3.75(t-1) + 3(t-2) - 11.25t + 33.75 = 0
t = 16/3 or 5 1/3 hrs, which would put it at 5:20 pm
I will let you check my work