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.c. What is an appropriate domain for ddd?

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 32c

1장, 문제 32c

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$)^2$ .
c. What is an appropriate domain for $d$ ?

검증된 단계별 안내

First, understand that the function d(t) = (5 - 0.22t)^2 represents the depth of water in the tank at time t. The goal is to find the appropriate domain for this function, which means determining the values of t for which the function is valid.

Since the depth of water cannot be negative, we need to find when d(t) becomes zero. Set the equation (5 - 0.22t)^2 = 0 and solve for t.

Take the square root of both sides to simplify the equation: 5 - 0.22t = 0.

Solve for t by isolating it on one side of the equation: 0.22t = 5.

Divide both sides by 0.22 to find the maximum value of t: t = 5 / 0.22. The domain of d(t) is from t = 0 to this value, as the depth is non-negative in this interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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, as it relates the height of the water to the velocity of the outflow.

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. In the context of this problem, they are used to model the rate of change of the water depth in the tank over time, allowing us to derive the function that describes the depth as a function of time.

Domain of a Function

The domain of a function refers to the set of all possible input values (in this case, time) for which the function is defined. For the depth function d(t), the domain must consider physical constraints, such as the time starting from when the drain is opened and the maximum depth of water in the tank, ensuring that the function remains valid throughout the draining process.