A horizontal cylindrical tank has cross-sectional area A(x)=4(6 x-x^2) m^2 at height x meters ab

step 1 This problem right here is going to be some related rates. We know that we want to find DX, DT,

A horizontal cylindrical tank has cross-sectional area...

A horizontal cylindrical tank has cross-sectional area $A(x)=4\left(6 x-x^{2}\right) m^{2}$ at height $x$ meters above the bottom when $x \leq 3$ . a. The volume $V$ between heights $a$ and $b$ is $\int_{a}^{b} A(x) d x$ . Find the volume at heights between 2 m and 3 $\mathrm{m}$ . b. Suppose that oil is being pumped into the tank $\frac{d x}{d t}=\frac{d x}{d V} \frac{d V}{d t},$ at how many meters per minute is the height of oil in the tank changing, expressed in terms of $x,$ when the height is at $x$ meters? c. How long does it take to fill the tank to 3 $\mathrm{m}$ starting from a fill level of 2 $\mathrm{m} ?$

A horizontal cylindrical tank has cross-sectional area $A(x)=4\left(6 x-x^{2}\right) m^{2}$ at height $x$ meters above the bottom when $x \leq 3$ .
a. The volume $V$ between heights $a$ and $b$ is $\int_{a}^{b} A(x) d x$ . Find the volume at heights between 2 m and 3 $\mathrm{m}$ .
b. Suppose that oil is being pumped into the tank $\frac{d x}{d t}=\frac{d x}{d V} \frac{d V}{d t},$ at how many meters per minute is the height of oil in the tank changing, expressed in terms of $x,$ when the height is at $x$ meters?
c. How long does it take to fill the tank to 3 $\mathrm{m}$ starting from a fill level of 2 $\mathrm{m} ?$

Key Concepts

Integration for Time Calculation

In systems where the rate of change of a variable is known in terms of another variable, integration can be used to determine the total time required for a process to occur. This involves setting up and solving an integral that relates the differential changes in one variable to the passage of time, thereby providing a method to calculate the total duration of an event.

Related Rates

Related rates problems connect the rates at which different quantities change over time. This concept is particularly useful in problems where a physical process involves interdependent variables, such as the height of a liquid in a tank changing as the volume changes with a constant inflow or outflow.

Definite Integration for Volume Calculation

This concept involves using the definite integral to compute the volume of a solid when given a cross-sectional area that varies with height. By integrating the area function over a specific interval, one can obtain the volume between those heights, which is fundamental in applications such as finding the amount of liquid in a container.

Chain Rule in Differentiation

The chain rule is a critical tool in calculus for differentiating composite functions. In situations where one quantity depends on another that in turn depends on a third (such as height depending on volume which depends on time), the chain rule allows us to systematically relate their rates of change.

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