36 calculus Let P(t) be the performance level of someone learning a skill as a f

Question

36 calculus

Let P(t) be the performance level of someone learning a skill as a function of the training time. The graph of P is called a learning curve. The following differential equation is proposed as a reasonable model for learning, where k is a positive constant. dP/dt = k(M - P(t)) Two new workers were hired for an assembly line. Jim processed 54 units during the first hour and 72 units during the second hour. Mark processed 44 units during the first hour and 72 units the second hour. Using this model and assuming that P(0) = 0, estimate the maximum number of units per hour that each worker is capable of processing.

Explanation / Answer

given

dP/dt=k(MP(t))

dP/(MP(t)) =k dt

dP/(M(1P(t)/M)) =kdt

dP/(1P(t)/M) =Mkdt

integrate on both sides

dP/(1P(t)/M) = Mkdt

-M(ln(1-P(t)/M))=Mkt +c

(ln(1-P(t)/M))=-kt +c

(1-P(t)/M)=e-kt +c

(1-P(t)/M)=Ce-kt

P(t)/M=1-Ce-kt

given P(0)=0

0/M=1-Ce-k0

0=1-C

C=1

P(t)/M=1-e-kt

P(t)=M(1-e-kt)

JIM could process 54 units per minute after one hour

54=M(1-e-k)

=>e-k=1-(54/M) ------------>(1)

72 units per minute after two hours

72=M(1-e-2k)

from (1)

72=M(1-(1-(54/M))2)

72=M(1-(1-(108/M) +(2916/M2))

72=M((108/M) -(2916/M2))

72=((108) -(2916/M))

2916/M=108-72

2916/M=36

M=2916/36

M=81

maximum number of units per minute that JIM is capable of processing. =81

==================================

P(t)=M(1-e-kt)

MARK could process 44 units per minute after one hour

44=M(1-e-k)

=>e-k=1-(44/M) ------------>(1)

72 units per minute after two hours

72=M(1-e-2k)

from (1)

72=M(1-(1-(44/M))2)

72=M(1-(1-(88/M) +(1936/M2))

72=M((88/M) -(1936/M2))

72=((88) -(1936/M))

1936/M=88-72

1936/M=16

M=1936/16

M=121

maximum number of units per minute that MARK is capable of processing. =121

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.