Q 84. Suppose the rate at which a popu... [FREE SOLUTION]

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Chapter 5: Q 84. (page 431)

Suppose the rate at which a population of $100$ weasels catches weasel-mumps is $r (t) = 8 - 2 \ln (t + 1)$ weasels per day, where $t > 0$ measures the number of days after the initial outbreak. Find a formula that represents the number $W (t)$ of weasels that have contracted weasel-mumps $t$ days after the initial outbreak. Approximately how long will it until all but one of the weasels are infected?

Short Answer

Expert verified

The solution is $1000 - 202 \ln (101)$ .

Step by step solution

Step 1. Given information.

The rate of weasel-mump infection is $r (t) = 8 - 2 \ln (t + 1)$ .

Step 2. Find a formula that represents the number W(t).

$\begin{array}{rcl} 鈭� r (t) & = & 鈭� \{8 - 2 \ln (t + 1)\} d t \\ = & 8 鈭� d t - 2 鈭� \ln (t + 1) d t \\ = & 8 t - 2 [\ln (t + 1) 鈭� d t - 鈭� \{\frac{d}{d t} (\ln (t + 1)) 鈭� d t\} d t] \\ = & 8 t - 2 [\ln (t + 1) 路 t - 鈭� \frac{1}{t + 1} 路 t d t] \\ = & 8 t - 2 t 路 \ln (t + 1) + 2 鈭� \{1 - \frac{1}{t + 1}\} d t \\ = & 8 t - 2 t 路 \ln (t + 1) + 2 鈭� d t + 2 鈭� \frac{1}{t + 1} d t \\ = & 8 t - 2 t 路 \ln (t + 1) + 2 t - 2 [鈭� \frac{1}{u} d u] 鈥� 鈥� 鈥� 鈥� 鈥� \{u = t + 1, d u = d t\} \\ = & 8 t - 2 t 路 \ln (t + 1) - 2 \ln |u| \\ = & 10 t - 2 t 路 \ln (t + 1) - 2 \ln |t + 1| \end{array}$

Step 3. Now calculate the time.

$\begin{array}{rcl} W (t) & = & 10 t - 2 t 路 \ln (t + 1) - 2 \ln |t + 1| \\ W (100) & = & 10 (100) - 2 (100) 路 \ln (100 + 1) - 2 \ln |100 + 1| \\ = & 1000 - 200 \ln (101) - 2 \ln (101) \\ = & 1000 - 202 \ln (101) \end{array}$

Step 4. Simplified answer.

Hence, the required value is $1000 - 202 \ln (101)$ .

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91影视

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find a formula that represents the number W(t).

Step 3. Now calculate the time.

Step 4. Simplified answer.

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