A cylindrical water reservoir is served with 3 pumps P,Q,R. Pump P and Q can fill the empty reservoir in 5 hours while pump Q can fill the same reservoir in 10 hours. When full pump R can drain the reservoir in 3 hours. a) One day the reservoir was empty and pumps P and Q opened at 8.45 am.At 11.15am pump P developed an electric fault. Pump Q continued operating alone till the reservoir was full. Find the time of the day that the reservoir was completely filled. b) On another day the reservoir was halfway filled with water. Pumps P and R were then opened at the same time . How long did it take to completely drain the reservoir. c) The reservoir's dimensions are base diameter 14 metres and height 12.5 meters. The water in (b) above is filled in a water tanker of capacity 12000 litres. How many trips does the tanker makes to drain the reservoir.
a) From 8.45 am to 11.15 am, there is a time span of 2.5 hours. During this time, pumps P and Q are operating together, so the rate at which they fill the reservoir is 1/5 + 1/10 = 3/10 of the reservoir per hour. Therefore, in 2.5 hours, they can fill 2.5 * (3/10) = 0.75 of the reservoir.
After pump P develops an electric fault, pump Q continues operating alone. We need to find out how much more time pump Q takes to fill the remaining 1 - 0.75 = 0.25 of the reservoir.
Since pump Q can fill the reservoir in 10 hours, it can fill 1/10 of the reservoir in 1 hour. Therefore, it takes 0.25 / (1/10) = <<0.25/(1/10)=2.5>>2.5 hours for pump Q to fill the remaining 0.25 of the reservoir.
Therefore, the total time taken for the reservoir to be completely filled is 2.5 + 2.5 = 5 hours.
The reservoir is completely filled at 11.15 am + 5 hours = 4.15 pm.
b) Since pump P and pump R are operating together, their combined drainage rate is 1/5 - 1/3 = (3 - 5)/15 = -2/15 of the reservoir per hour. This negative rate represents draining the reservoir.
Since the reservoir is halfway filled, it means there is 0.5 of the reservoir remaining to be drained. To find out how long it takes to completely drain the reservoir, we need to find the time it takes for pump P and pump R to drain 0.5 of the reservoir.
The rate at which pump P and pump R can drain the reservoir is -2/15 of the reservoir per hour. Therefore, it takes 0.5 / (-2/15) = -(0.5 * 15/2) = -3.75 hours to drain 0.5 of the reservoir.
The negative sign means that it takes 3.75 hours before the reservoir is completely drained.
c) The volume of the reservoir can be calculated using the formula for the volume of a cylinder:
Volume = π * r^2 * h
where r is the radius of the base (which is half of the diameter), and h is the height of the cylinder.
The radius of the base is 14 / 2 = 7 m.
Volume = π * (7^2) * 12.5 = 1372.79 m^3
Since 1 m^3 is equal to 1000 liters, the capacity of the reservoir is 1372.79 * 1000 = 1,372,790 liters.
Since the water tanker has a capacity of 12,000 liters, the number of trips it makes to drain the reservoir is 1,372,790 / 12,000 = 114.4.
Therefore, the water tanker makes 115 trips to drain the reservoir.