a swimming pool is 40m long and 12m wide with a shallow end and a deep end.The shallow end is 1.3m deep and 10m wide while the deep end is 3m deep and 10m wide. The center tappers gently from the shallow end to the deep end. The pool is filled by four similar pipes each of diameter 7cm. Water flows through each pipe at the rate of 1.2 meters per second. Calculate to the nearest minute the time it takes to fill the pool
You say the pool is 12m wide, but both ends are only 10m wide. I'll assume that's a typo, and the pool is 10m wide.
The side view is a trapezoid with bases of 1.3 and 3, and an altitude of 40. So the volume of the pool is
(1.3+3)/2 * 40 * 10 = 860 m^3
The volume of water coming in is 4 * π * (7/2)^2 * 1200 = 58800π cm^3/s = 0.1847 m^3/s
So it will take
860/0.1847 * 1/60 = 77.59 minutes
to fill the pool.
Adjust the numbers if the width of the pool is in fact not 10 m.
To calculate the time it takes to fill the pool, we need to find the volume of the pool first.
Let's calculate the volume of the shallow end:
Length of the shallow end = 40m
Width of the shallow end = 10m
Depth of the shallow end = 1.3m
Volume of the shallow end = Length x Width x Depth
Volume of the shallow end = 40m x 10m x 1.3m
Volume of the shallow end = 520 cubic meters
Now let's calculate the volume of the deep end:
Length of the deep end = 40m
Width of the deep end = 10m
Depth of the deep end = 3m
Volume of the deep end = Length x Width x Depth
Volume of the deep end = 40m x 10m x 3m
Volume of the deep end = 1200 cubic meters
Now, let's calculate the average depth of the pool. Since the pool tapers gently from the shallow end to the deep end, we can calculate the average depth as the average of the shallow and deep end depths:
Average depth = (Shallow end depth + Deep end depth) / 2
Average depth = (1.3m + 3m) / 2
Average depth = 2.15m
Now, let's calculate the volume of the pool:
Volume of the pool = Volume of the shallow end + Volume of the deep end
Volume of the pool = 520 cubic meters + 1200 cubic meters
Volume of the pool = 1720 cubic meters
Since the water flows through four similar pipes and each pipe has a diameter of 7cm, we can calculate the flow rate of a single pipe.
Radius of the pipe = Diameter / 2 = 7cm / 2 = 3.5cm
Now, let's convert the radius to meters:
Radius = 3.5cm = 3.5 / 100 = 0.035m
Area of the pipe = π x (Radius)^2
Area of the pipe = π x (0.035m)^2
Now, let's calculate the flow rate of a single pipe:
Flow rate of a single pipe = Area of the pipe x Speed of water
Flow rate of a single pipe = π x (0.035m)^2 x 1.2 m/s
Total flow rate of all four pipes = 4 x Flow rate of a single pipe
Now, let's calculate the time it takes to fill the pool:
Time to fill the pool = Volume of the pool / Total flow rate of all four pipes
Time to fill the pool = 1720 cubic meters / (4 x Flow rate of a single pipe)
Now, plug in the values and calculate the time:
Time to fill the pool = 1720 cubic meters / (4 x π x (0.035m)^2 x 1.2 m/s)
Calculating this expression gives us the time it takes to fill the pool in seconds. To convert it to minutes, divide by 60:
Time to fill the pool (in minutes) = Time to fill the pool (in seconds) / 60