$\"\"$

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The maximum number of units a worker can produce in a day is $\"\"$ .

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(a)

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The rate of increase in the number of units $\"\"$ produced with respect to time $\"\"$ in days by a new employee is proportional to $\"\"$ is $\"\"$ .

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$\"\"$ .

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The differential equation for the rate of change of performance with respect to time $\"\"$ is $\"\"$ . $\"\"$

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(b) Solve the differential equation.

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The differential equation is $\"\"$ . $\"\"$ .

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The equation is in the form of $\"\"$ .

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The differential equation is a first order linear differential equation.

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Solution of the first order linear differential equation is $\"\"$ is $\"\"$ .

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

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Find $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ . $\"\"$

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The solution of differential equation is $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Therefore, the equation is $\"\"$ .

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$\"\"$

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(c)

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The equation is $\"\"$ .

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On the first day a new employee produced $\"\"$ .

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Hence, $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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$\"\"$ .

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$\"\"$

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The equation is $\"\"$ .

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On the twentieth day new employee produced $\"\"$ .

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Hence, $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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$\"\"$ .

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Therefore, the solution is $\"\"$ .

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$\"\"$

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(a) The differential equation is $\"\"$ .

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(b) The function is $\"\"$ .

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(c) The solution is $\"\"$ .