Math Problem Statement

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(1 − ekt). After 20 days on the job, a new employee produces 16 units. (a) Find the learning curve for this employee (first, find the value of k). (Round your answer to three decimal places.) k = N =

(b) How many days should pass before this employee is producing 23 units per day? (Round your answer to the nearest whole number.) days

Solution

We are given the learning curve for the number of units NN produced per day after a new employee has worked tt days, modeled by:

N=30(1ekt)N = 30(1 - e^{kt})

We also know that after 20 days, the employee produces 16 units. We can use this information to find the value of kk.

Part (a) - Find the value of kk

We can substitute N=16N = 16 and t=20t = 20 into the equation:

16=30(1e20k)16 = 30(1 - e^{20k})

Now, let's solve for kk.

  1. Divide both sides of the equation by 30: 1630=1e20k\frac{16}{30} = 1 - e^{20k}

1630=0.5333\frac{16}{30} = 0.5333

0.5333=1e20k0.5333 = 1 - e^{20k}

  1. Subtract 0.5333 from 1: e20k=0.4667e^{20k} = 0.4667

  2. Take the natural logarithm of both sides: 20k=ln(0.4667)20k = \ln(0.4667)

  3. Solve for kk: k=ln(0.4667)20k = \frac{\ln(0.4667)}{20}

Let's calculate kk.The value of kk is approximately 0.038-0.038 (rounded to three decimal places).

Thus, the learning curve equation becomes:

N=30(1e0.038t)N = 30(1 - e^{-0.038t})

Part (b) - How many days should pass before this employee produces 23 units per day?

We need to find tt when N=23N = 23. Using the equation N=30(1e0.038t)N = 30(1 - e^{-0.038t}), substitute N=23N = 23:

23=30(1e0.038t)23 = 30(1 - e^{-0.038t})

Let's solve for tt.It appears that the result for tt is negative, which is not physically meaningful in this context. This suggests a mistake in the interpretation of the model. Let me correct and adjust for the proper solution approach for part (b).It will take approximately 38 days for the employee to produce 23 units per day.

Summary:

  • k0.038k \approx -0.038
  • Learning curve equation: N=30(1e0.038t)N = 30(1 - e^{-0.038t})
  • The employee will produce 23 units per day after about 38 days.

Would you like more details on any step or have any further questions?

Here are 5 related questions to consider:

  1. How would the learning curve change if the maximum number of units was higher?
  2. What does the parameter kk represent in the learning curve equation?
  3. How would the model change if the rate of learning was slower?
  4. What is the significance of the constant 30 in this equation?
  5. How can we modify this model to account for workers who may regress in productivity?

Tip: In learning curve models, kk represents the rate of improvement. A more negative kk indicates faster learning.

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Math Problem Analysis

Mathematical Concepts

Exponential Growth and Decay
Learning Curves
Algebra

Formulas

N = 30(1 − e^(kt))
k = ln(0.4667) / 20

Theorems

Exponential Decay Theorem

Suitable Grade Level

Grades 11-12