Two identical cylindrical tanks, A and B, are being filled with water - Leaving Cert Mathematics - Question Question 2 - 2013
Question Question 2
Two identical cylindrical tanks, A and B, are being filled with water. At a particular time, the water in tank A is 25 cm deep and the depth of the water is increasi... show full transcript
Worked Solution & Example Answer:Two identical cylindrical tanks, A and B, are being filled with water - Leaving Cert Mathematics - Question Question 2 - 2013
Step 1
Each tank is 1 m in height. Find how long it takes to fill each tank.
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Answer
To calculate the time it takes to fill each tank, we need to determine the total volume of water to fill each tank and then divide it by the rate of water flow.
For Tank A:
Initial Depth: 25 cm
Filling Rate: 5 cm per 10 seconds, equivalent to 30 cm/min
Total Height: 100 cm
Remaining Depth to Fill: 100 cm - 25 cm = 75 cm
Time to fill Tank A:
Time = Remaining Depth / Filling Rate = 75 cm / (5 cm/10 s) = 150 seconds = 2 min 30 s.
For Tank B:
Initial Depth: 10 cm
Filling Rate: 7.5 cm per 10 seconds, equivalent to 45 cm/min
Total Height: 100 cm
Remaining Depth to Fill: 100 cm - 10 cm = 90 cm
Time to fill Tank B:
Time = Remaining Depth / Filling Rate = 90 cm / (7.5 cm/10 s) = 120 seconds = 2 minutes.
Step 2
For each tank, write down a formula which gives the depth of water in the tank at any given time. State clearly the meaning of any letters used in your formulas.
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Answer
The depth of water in each tank as a function of time can be expressed using the following formulas:
Tank A:d=25+105t where:
d = depth in cm at time t in seconds.
Tank B:d=10+107.5t where:
d = depth in cm at time t in seconds.
Step 3
Find, from your graphs, how much time passes before the depth of water is the same in each tank.
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Answer
From the graph, it can be observed that the depth of water in both tanks becomes equal after 60 seconds when both tanks are at a depth of 25 cm.
Step 4
Verify your answer to part (e) using your formulas from part (e).
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Answer
To verify, set the two depth equations equal to each other:
25+105t=10+107.5t
Solving for t yields:
Bringing like terms together:
15=107.5t−105t
Combining terms gives:
15=102.5t
Solving for t results in:
t=2.515×10=60 seconds.
Thus, the solution is verified.
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