If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as
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V 5000(1 40 t) 0 t 40
Find the rate at which water is draining from the tank after
(a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings.
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If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as
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Calculus with Analytic Geometry I Exam 6, Friday ... - FAU Math
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In that pdf document question 2 is your question.
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To find the rate at which water is draining from the tank after a given time, we need to find the derivative of the volume function V(t) with respect to time t.
The given volume function is:
V(t) = 5000(1 - 40t/40)
Simplifying this equation, we have:
V(t) = 5000(1 - t)
To find the rate at which water is draining from the tank, we need to find the derivative dV(t)/dt.
dV(t)/dt = 5000 * (-1) = -5000
The negative sign indicates that the drainage rate is decreasing. So, the water is draining at a rate of 5000 gallons per minute.
Now, let's find the rate at which water is draining from the tank after different time intervals:
(a) After 5 minutes:
We substitute t = 5 into the derivative equation:
dV(5)/dt = -5000
The water is draining at a rate of -5000 gallons per minute after 5 minutes.
(b) After 10 minutes:
We substitute t = 10 into the derivative equation:
dV(10)/dt = -5000
The water is still draining at a rate of -5000 gallons per minute after 10 minutes.
(c) After 20 minutes:
We substitute t = 20 into the derivative equation:
dV(20)/dt = -5000
The water is still draining at a rate of -5000 gallons per minute after 20 minutes.
(d) After 40 minutes:
We substitute t = 40 into the derivative equation:
dV(40)/dt = -5000
Again, the water is draining at a rate of -5000 gallons per minute after 40 minutes.
The water flows out of the tank at a constant rate of -5000 gallons per minute throughout the entire 40-minute period. The drainage rate does not change over time, indicating that the fastest and slowest drainage rates occur at the same time, which is 40 minutes.
In summary, the rate at which the water is draining from the tank is a constant -5000 gallons per minute. The water flows out at the same rate regardless of the time elapsed.