$\"\"$

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Length of the trough $\"\"$ m.

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Height of the isosceles trapezoid is $\"\"$ m.

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Width of the trapezoid at the bottom is $\"\"$ cm.

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Trough is being filled with water at a rate of $\"\"$ m³/min.

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Thus, $\"\"$ .

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Diagram of the situation at any time $\"\"$ :

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$\"\"$

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Consider at any time $\"\"$ , the width of the water level is $\"\"$ and height of the water level is $\"\"$ .

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Consider the triangle in above figure.

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$\"\"$

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From the property of the similar triangles.

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$\"\"$

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Volume of water consists two parts, one is rectangular solid and two solid prisms.

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Volume of the rectangular solid is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in above expression.

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$\"\"$

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Volume of the prism is $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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$\"\"$

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Volume of the two triangular prisms is

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$\"\"$

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Volume of the entire water level in trough is sum of the rectangular and triangular prism volumes.

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$\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Find the rate at which water level is rising when it is $\"\"$ inches deep.

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Thus, $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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Rate at which water level rising is $\"\"$ cm/min.

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$\"\"$

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Rate at which water level rising is $\"\"$ cm/min.