Chapter 5.4, Problem 126AYU

(a)

To determine

The rate of learning $k$ , if the student has learned $20$ vocabulary words after $5$ minutes, where psychologists sometimes use the function $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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

  $A=200$

  $\Rightarrow 20=200\left(1-e^{-k\left(5\right)}\right)$

Divide both sides by $200$

  $\Rightarrow \frac{20}{200}=1-e^{-5k}$

  $\Rightarrow \frac{1}{10}=1-e^{-5k}$

Subtract $1$ from both sides,

  $\Rightarrow -\frac{9}{10}=-e^{-5k}$

  $\Rightarrow \frac{9}{10}=e^{-5k}$

Taking natural log on both sides,

  $\Rightarrow \ln \left(\frac{9}{10}\right)=\ln \left(e^{-5k}\right)$

  $\Rightarrow \ln \left(\frac{9}{10}\right)=-5k\ln \left(e\right)$

  $\Rightarrow \ln \left(\frac{9}{10}\right)=-5k$

Divide both sides by $-5$ ,

  $\Rightarrow -\frac{\ln \left(0.9\right)}{5}=k$

  $\Rightarrow k=0.021$

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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

  $\Rightarrow L\left(t\right)=200\left(1-e^{-0.021t}\right)$

To find the number of words the student has learned after $10$ minutes

  $\Rightarrow L\left(t\right)=200\left(1-e^{-0.021\left(10\right)}\right)$

By simplifying,

  $\Rightarrow L\left(t\right)=200\left(1-e^{-0.21}\right)$

  $=200-200e^{-0.21}$

  $=37.88\approx 38$

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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

  $\Rightarrow L\left(t\right)=200\left(1-e^{-0.021t}\right)$

To find the number of words the student has learned after $15$ minutes,

  $\Rightarrow L\left(t\right)=200\left(1-e^{-0.021\left(15\right)}\right)$

By simplifying,

  $L\left(t\right)=200\left(1-e^{-0.315}\right)$

  $=200-200e^{-0.315}$

  $=54.04222\approx 54$

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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

Answer to Problem 126AYU

Solution:

The student takes approximately $110$minutes to learn $180$ vocabulary words.

Explanation of Solution

Given information:

Psychologists sometimes use the function $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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 $L\left(t\right)=A\left(1-e^{-kt}\right)$ 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.

  $\Rightarrow L\left(t\right)=200\left(1-e^{-0.021t}\right)$

To find time for the student to learn $180$ vocabulary words,

  $\Rightarrow 180=200\left(1-e^{-0.021t}\right)$

Divide both sides by $200$ ,

  $\Rightarrow \frac{180}{200}=1-e^{-0.021t}$

  $\Rightarrow \frac{9}{10}=1-e^{-0.021t}$

Subtract $1$ from both sides,

  $\Rightarrow \frac{9}{10}-1=1-e^{-0.021t}-1$

  $\Rightarrow -\frac{1}{10}=-e^{-0.021t}$

  $\Rightarrow \frac{1}{10}=e^{-0.021t}$

Taking natural log on both sides,

  $\Rightarrow \ln \left(0.1\right)=\ln \left(e^{-0.021t}\right)$

By using $\ln \left(a^b\right)=b\ln a$ ,

  $\Rightarrow \ln \left(0.1\right)=-0.021t\ln \left(e\right)$

  $\Rightarrow \ln \left(0.1\right)=-0.021t$

Divide both sides by $-0.021$

  $\Rightarrow \frac{\ln \left(0.1\right)}{-0.021}=t$

  $\Rightarrow t=109.6469\approx 110$ minutes

Therefore, the student takes approximately $110$ minutes to learn $180$ vocabulary words.