Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of 101010 m2 is filled to a depth of 252525 m with water. At t=0t=0t=0 s, a drain in the bottom of the tank with an area of 111 m² is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time t≥0t\geq{0}t≥0 is d(t)=(5−0.22t)2d\left(t\right)=\left(5-0.22t\$$right)^2d(t)=(5$$−0.22t)2. a. Check that d(0)=25d\left(0\right)=25d(0)=25, as specified.

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 32a

1장, 문제 32a

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m² is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t$\geq{0}$$ is $d$\left$(t$\right$)=$\left$(5-0.22t$\right$)^2$ .
a. Check that $d$\left$(0$\right$)=25$ , as specified.

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Identify the function given for the depth of water in the tank: $ d(t) = (5 - 0.22t)^2 $.

To verify $ d(0) = 25 $, substitute $ t = 0 $ into the function $ d(t) $.

Calculate $ d(0) = (5 - 0.22 \times 0)^2 $.

Simplify the expression: $ d(0) = (5 - 0)^2 = 5^2 $.

Compute $ 5^2 $ to confirm that $ d(0) = 25 $, which matches the specified initial condition.

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주요 개념

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Torricelli's Law

Torricelli's Law states that the speed of fluid flowing out of an orifice under the force of gravity is proportional to the square root of the height of the fluid above the opening. This principle is crucial for understanding how the depth of water in a tank changes over time as it drains. The law can be mathematically expressed as v = √(2gh), where v is the exit speed, g is the acceleration due to gravity, and h is the height of the fluid.

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. In the context of draining a tank, they are used to model the rate of change of the water depth over time. The equation derived from Torricelli's Law can be expressed as a first-order differential equation, which can be solved to find the function describing the depth of water as a function of time.

Initial Conditions

Initial conditions are the values that specify the state of a system at the beginning of a process. In this problem, the initial condition is that the depth of water in the tank at time t=0 is 25 meters. This information is essential for solving the differential equation, as it allows for the determination of the specific solution that describes the water depth over time.