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The image contains a problem involving a cylindrical tank and some algebraic equations. I'll transcribe and solve it below:

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**Problem:**

A cylindrical tank with radius 4 feet and a height of 10 feet is being filled by water from a hose that has a flow rate of 20 ft³/min. Let \( y \) represent the water height in the tank and \( t \) be the time (in minutes).

The following three versions of the problem can be assigned:

1. Find \(\frac{dy}{dt}\) when \( y = 5 \) feet.

Given:
\[ V = \pi r^2 y \]
\[ \frac{dV}{dt} = 20 \text{ ft}^3/\text{min} \]

Let's solve for \(\frac{dy}{dt}\):

Given \( r = 4 \) ft, the volume \( V \) of water in the tank as a function of height \( y \) is:
\[ V = \pi (4^2) y = 16\pi y \]

Differentiate \( V \) with respect to \( t \):
\[ \frac{dV}{dt} = 16\pi \frac{dy}{dt} \]

We know \( \frac{dV}{dt} = 20 \) ft³/min, so:
\[ 20 = 16\pi \frac{dy}{dt} \]

Solving for \(\frac{dy}{dt}\):
\[ \frac{dy}{dt} = \frac{20}{16\pi} = \frac{5}{4\pi} \]

Hence,
\[ \frac{dy}{dt} \approx \frac{5}{4\pi} \approx 0.3978 \text{ ft/min} \]

2. Find the original height of the tank and show the intermediate steps. (Assume the answer is in correct range).

Given:
\[ h = 10 \text{ feet} \]

The tank height is already given as 10 feet.

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Do you have any questions or need further details on any part of this solution?

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**Related Questions:**

1. How do you find the volume of a cylinder given its radius and height?
2. What is the formula for the volume of a cylinder in terms of its radius and height?
3. How do you differentiate the volume of a cylinder with respect to time?
4. How can you calculate the flow rate needed to fill a tank in a certain time?
5. What happens to \(\frac{dy}{dt}\) if the radius of the cylinder changes?
6. How would the problem change if the flow rate were different?
7. What is the significance of \(\frac{dy}{dt}\) in this context?
8. How would you integrate \(\frac{dy}{dt}\) to find the height of the water at any given time?

**Tip:** Remember that the chain rule is essential when differentiating composite functions in problems involving rates of change.

Learn how to solve the rate of change problem involving a cylindrical tank being filled with water. Includes detailed steps for finding dy/dt given the tank's dimensions and water flow rate. Suitable for advanced high school or college-level students studying calculus and rate of change problems.

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