The rate of learning k , if the student has learned 20 vocabulary words after 5 minutes, where psychologists sometimes use the function L ( t ) = A ( 1 − e − k t ) to measure the amount L learned at time t . Here, A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
The rate of learning k , if the student has learned 20 vocabulary words after 5 minutes, where psychologists sometimes use the function L ( t ) = A ( 1 − e − k t ) to measure the amount L learned at time t . Here, A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
The rate of learning k , if the student has learned 20 vocabulary words after 5 minutes, where psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t . Here, A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
(a)
Expert Solution
Answer to Problem 126AYU
Solution:
The rate of learning k=0.021 vocabulary words per minutes.
Explanation of Solution
Given information:
Psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t .
Here, A represents the amount to be learned and the number k measures the rate of learning.
Suppose that a student has an amount A of 200 vocabulary words to learn.
A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
The function L(t)=A(1−e−kt) to measure the amount L learned at time t .
The student has learned 20 vocabulary words after 5 minutes.
Hence, t=5,L=20
From the given information
A=200
⇒20=200(1−e−k(5))
Divide both sides by 200
⇒20200=1−e−5k
⇒110=1−e−5k
Subtract 1 from both sides,
⇒−910=−e−5k
⇒910=e−5k
Taking natural log on both sides,
⇒ln(910)=ln(e−5k)
⇒ln(910)=−5kln(e)
⇒ln(910)=−5k
Divide both sides by −5 ,
⇒−ln(0.9)5=k
⇒k=0.021
Therefore, k=0.021 vocabulary words per minutes
(b)
To determine
The number of words the student will learn after 10 minutes, where psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t . Here, A represents the amount to be learned and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
(b)
Expert Solution
Answer to Problem 126AYU
Solution:
The number of words the student will learn after 10 minutes is approximately 38
Explanation of Solution
Given information:
Psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t .
Here, A represents the amount to be learned and the number k measures the rate of learning.
Suppose that a student has an amount A of 200 vocabulary words to learn.
A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
The function L(t)=A(1−e−kt) to measure the amount L learned at time t .
From the given information, A=200 vocabulary words
From part (a), k=0.021 vocabulary words per minutes.
⇒L(t)=200(1−e−0.021t)
To find the number of words the student has learned after 10 minutes
⇒L(t)=200(1−e−0.021(10))
By simplifying,
⇒L(t)=200(1−e−0.21)
=200−200e−0.21
=37.88≈38
Therefore, the student will learn 38 vocabulary words after 10 minutes.
(c)
To determine
The number of words the student will earn after 15 minutes, where psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t . Here, A represents the amount to be learned and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
(c)
Expert Solution
Answer to Problem 126AYU
Solution:
The number of words the student will learn after 15 minutes is approximately 54 .
Explanation of Solution
Given information:
Psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t .
Here, A represents the amount to be learned and the number k measures the rate of learning.
Suppose that a student has an amount A of 200 vocabulary words to learn.
A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
The function L(t)=A(1−e−kt) to measure the amount L learned at time t .
From the given information, A=200 vocabulary words,
From part (a), k=0.021 vocabulary words per minutes.
⇒L(t)=200(1−e−0.021t)
To find the number of words the student has learned after 15 minutes,
⇒L(t)=200(1−e−0.021(15))
By simplifying,
L(t)=200(1−e−0.315)
=200−200e−0.315
=54.04222≈54
Therefore, the student will learn approximately 54 vocabulary words after 15 minutes.
(d)
To determine
The time for the student to learn 180 vocabulary words, where psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t . Here, A represents the amount to be learned and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
(d)
Expert Solution
Answer to Problem 126AYU
Solution:
The student takes approximately 110minutes to learn 180 vocabulary words.
Explanation of Solution
Given information:
Psychologists sometimes use the function L(t)=A(1−e−kt) to measure the amount L learned at time t .
Here, A represents the amount to be learned and the number k measures the rate of learning.
Suppose that a student has an amount A of 200 vocabulary words to learn.
A psychologist determines that the student has learned 20 vocabulary words after 5 minutes.
The function L(t)=A(1−e−kt) to measure the amount L learned at time t .
From the given information A=200 vocabulary words
From part (a), k=0.021 vocabulary words per minutes.
⇒L(t)=200(1−e−0.021t)
To find time for the student to learn 180 vocabulary words,
⇒180=200(1−e−0.021t)
Divide both sides by 200 ,
⇒180200=1−e−0.021t
⇒910=1−e−0.021t
Subtract 1 from both sides,
⇒910−1=1−e−0.021t−1
⇒−110=−e−0.021t
⇒110=e−0.021t
Taking natural log on both sides,
⇒ln(0.1)=ln(e−0.021t)
By using ln(ab)=blna ,
⇒ln(0.1)=−0.021tln(e)
⇒ln(0.1)=−0.021t
Divide both sides by −0.021
⇒ln(0.1)−0.021=t
⇒t=109.6469≈110 minutes
Therefore, the student takes approximately 110 minutes to learn 180 vocabulary words.
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