If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then Torricelli's Law gives the volume [tex]V[/tex] of water remaining in the tank after [tex]t[/tex] minutes as
[tex]V=5000\left(1-\frac{t}{40}\right)^2[/tex]
Find the rate at which water is draining from the tank after 6 minutes. Round your answer to the nearest tenth.
The rate is [$\square$] gallons per minute.
Answer (2)
Find the derivative of the volume function V ( t ) = 5000 ( 1 − 40 t ) 2 with respect to time t , which gives d t d V = − 250 ( 1 − 40 t ) .
Evaluate the derivative at t = 6 minutes: d t d V ( 6 ) = − 250 ( 1 − 40 6 ) = − 212.5 .
Take the absolute value to find the rate of draining: ∣ − 212.5∣ = 212.5 .
The rate at which water is draining from the tank after 6 minutes is 212.5 gallons per minute.
Explanation
Problem Setup We are given the volume of water remaining in the tank after t minutes as V = 5000\[1 - \frac{t}{40}\]^2 We need to find the rate at which water is draining from the tank after 6 minutes. This rate is the absolute value of the derivative of the volume function with respect to time, evaluated at t = 6 .
Finding the Derivative First, we find the derivative of V with respect to t :
d t d V = 5000 ⋅ 2 ⋅ ( 1 − 40 t ) ⋅ ( − 40 1 ) = − 40 10000 ( 1 − 40 t ) = − 250 ( 1 − 40 t )
Evaluating the Derivative at t=6 Next, we evaluate the derivative at t = 6 :
d t d V ( 6 ) = − 250 ( 1 − 40 6 ) = − 250 ( 1 − 20 3 ) = − 250 ( 20 17 ) = − 25 ⋅ 2 17 = − 2 425 = − 212.5 The rate at which water is draining is the absolute value of this result.
Finding the Rate of Draining The rate at which water is draining from the tank after 6 minutes is ∣ − 212.5∣ = 212.5 gallons per minute.
Examples
Understanding rates of change is crucial in many real-world applications. For instance, in chemical engineering, it helps determine reaction rates; in economics, it's used to analyze growth rates; and in environmental science, it assists in modeling pollution dispersal. In this problem, we applied calculus to find the rate at which water drains from a tank, which is essential for managing water resources and designing efficient drainage systems. By knowing how quickly a tank empties, engineers can optimize the design of tanks and drainage systems for various applications, ensuring efficient and safe operations.
The rate at which water is draining from the tank after 6 minutes is 212.5 gallons per minute. This value is derived from the derivative of the volume function concerning time. After calculating and evaluating the derivative at t=6 minutes, the absolute value gives the final draining rate.
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