/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 The management at a plastics fac... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 56

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?

Short Answer

Expert verified
The value of \(k\) for this employee based on the given information and equation is determined. After a specific number of days, calculated using the value of \(k\), the employee is capable of producing 25 units in a day.

Step by step solution

01

Identify Given Variables

In the first sub-question, it is given that on day \(t = 20\), the employee produces \(N = 19\) units.
02

Substitute Given Variables in the Equation to Find \(k\)

Substitute \(N = 19\) and \(t = 20\) into the learning curve equation \(N = 30(1 - e^{kt})\). By doing this, we get 19 = 30(1 - e^{20k}). Solve this equation for \(k\).
03

Solve for \(k\)

To solve for \(k\), first, rearrange the equation to get \(e^{20k}\) on one side of the equation and a numerical value on the other side. Then, take the natural logarithm (ln) of both sides to eliminate \(e\), and finally solve for \(k\).
04

Substitute \(k\) into the Equation to Find \(t\)

In the second sub-question, it is asked to find the number of days \(t\) it will take to produce 25 units. Substitute \(N = 25\) and the value of \(k\) obtained from Step 3 into the equation. Rearrange the equation to get \(e^{kt}\) on one side of the equation and a numerical value on the other side. Then, take the natural logarithm (ln) of both sides to isolate \(t\). Finally, solve for \(t\).

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