Math / Algebra

QuestionAttempt 1: 10 attempts remaining. The town of Sickville, with a population of 9762 is exposed to the Blue Moon Virus, against which there is no immunity. The number of people infected when the virus is detected is 60 . Suppose the number of infections grows logistically (see logistic formula in question 4 above), with $k=0.84$ .
Find $b$ . $\square$ Find the formula for the number of people infected after $t$ days. $N(t)=$ $\square$ Find the number of people infected after 21 days. $\square$
Submit answer Next item

Studdy Solution

STEP 1

What is this asking? We need to figure out how many people in Sickville will catch the Blue Moon Virus after a certain number of days, knowing how many people are sick right now and how quickly the virus is spreading! Watch out! We're dealing with a logistic growth, not just exponential growth, so we need to account for the town's population limit.
Don't let the virus spread beyond Sickville's borders!

STEP 2

1. Find b
2. Define the function
3. Calculate the result

STEP 3

Remember, the logistic growth formula is $N(t) = \frac{P}{1 + be^{-kt}}$ , where $P$ is the total population, $k$ is the growth rate, and $N(0)$ is the initial number of infected people.

STEP 4

We know $P = \textbf{9762}$ , $k = \textbf{0.84}$ , and $N(0) = \textbf{60}$ .
Let's plug these values into our formula at $t=0$ : $N(0) = \frac{9762}{1 + be^{-0.84 \cdot 0}}$ .

STEP 5

Since $e^0 = 1$ , we get $60 = \frac{9762}{1 + b}$ .
Now, we want to isolate b.
Multiply both sides by $1+b$ to get $60(1+b) = 9762$ .
Divide both sides by 60: $1 + b = \frac{9762}{60} = \textbf{162.7}$ .
Finally, subtract 1 from both sides to find $b = 162.7 - 1 = \textbf{161.7}$ .

STEP 6

Now that we know $b = \textbf{161.7}$ , we can write the complete formula for the number of infected people after $t$ days: $N(t) = \frac{9762}{1 + 161.7e^{-0.84t}}$ .

STEP 7

We want to find the number of infected people after 21 days, so we plug $t = \textbf{21}$ into our formula: $N(21) = \frac{9762}{1 + 161.7e^{-0.84 \cdot 21}}$ .

STEP 8

First, calculate the exponent: $-0.84 \cdot 21 = \textbf{-17.64}$ .
Then, $e^{-17.64} \approx \textbf{2.29} \cdot 10^{-8}$ , which is a very small number!

STEP 9

Now, calculate the denominator: $1 + 161.7 \cdot (2.29 \cdot 10^{-8}) \approx 1 + \textbf{3.7} \cdot 10^{-6} \approx 1$ .

STEP 10

Finally, divide the numerator by the denominator: $N(21) = \frac{9762}{1} \approx \textbf{9762}$ .

STEP 11

After 21 days, approximately all 9762 people in Sickville will be infected with the Blue Moon Virus. $b$ is 161.7 and the formula is $N(t) = \frac{9762}{1 + 161.7e^{-0.84t}}$ .

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Find b
2. Define the function
3. Calculate the result

STEP 3

Remember, the logistic growth formula is $N(t) = \frac{P}{1 + be^{-kt}}$ , where $P$ is the total population, $k$ is the growth rate, and $N(0)$ is the initial number of infected people.

STEP 4

We know $P = \textbf{9762}$ , $k = \textbf{0.84}$ , and $N(0) = \textbf{60}$ .
Let's plug these values into our formula at $t=0$ : $N(0) = \frac{9762}{1 + be^{-0.84 \cdot 0}}$ .

STEP 5

Since $e^0 = 1$ , we get $60 = \frac{9762}{1 + b}$ .
Now, we want to isolate b.
Multiply both sides by $1+b$ to get $60(1+b) = 9762$ .
Divide both sides by 60: $1 + b = \frac{9762}{60} = \textbf{162.7}$ .
Finally, subtract 1 from both sides to find $b = 162.7 - 1 = \textbf{161.7}$ .

STEP 6

Now that we know $b = \textbf{161.7}$ , we can write the complete formula for the number of infected people after $t$ days: $N(t) = \frac{9762}{1 + 161.7e^{-0.84t}}$ .

STEP 7

We want to find the number of infected people after 21 days, so we plug $t = \textbf{21}$ into our formula: $N(21) = \frac{9762}{1 + 161.7e^{-0.84 \cdot 21}}$ .

STEP 8

First, calculate the exponent: $-0.84 \cdot 21 = \textbf{-17.64}$ .
Then, $e^{-17.64} \approx \textbf{2.29} \cdot 10^{-8}$ , which is a very small number!

STEP 9

Now, calculate the denominator: $1 + 161.7 \cdot (2.29 \cdot 10^{-8}) \approx 1 + \textbf{3.7} \cdot 10^{-6} \approx 1$ .

STEP 10

Finally, divide the numerator by the denominator: $N(21) = \frac{9762}{1} \approx \textbf{9762}$ .

STEP 11

After 21 days, approximately all 9762 people in Sickville will be infected with the Blue Moon Virus. $b$ is 161.7 and the formula is $N(t) = \frac{9762}{1 + 161.7e^{-0.84t}}$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Find *b*2. Define the function3. Calculate the result

STEP 3

Remember, the logistic growth formula is N(t)=P1+be−ktN(t) = \frac{P}{1 + be^{-kt}}N(t)=1+be−ktP​, where PPP is the **total population**, kkk is the **growth rate**, and N(0)N(0)N(0) is the **initial number of infected people**.

STEP 4

We know P=9762P = \textbf{9762}P=9762, k=0.84k = \textbf{0.84}k=0.84, and N(0)=60N(0) = \textbf{60}N(0)=60.Let's plug these values into our formula at t=0t=0t=0: N(0)=97621+be−0.84⋅0N(0) = \frac{9762}{1 + be^{-0.84 \cdot 0}}N(0)=1+be−0.84⋅09762​.

STEP 5

STEP 6

Now that we know b=161.7b = \textbf{161.7}b=161.7, we can write the complete formula for the number of infected people after ttt days: N(t)=97621+161.7e−0.84tN(t) = \frac{9762}{1 + 161.7e^{-0.84t}}N(t)=1+161.7e−0.84t9762​.

STEP 7

We want to find the number of infected people after **21** days, so we plug t=21t = \textbf{21}t=21 into our formula: N(21)=97621+161.7e−0.84⋅21N(21) = \frac{9762}{1 + 161.7e^{-0.84 \cdot 21}}N(21)=1+161.7e−0.84⋅219762​.

STEP 8

First, calculate the exponent: −0.84⋅21=-17.64-0.84 \cdot 21 = \textbf{-17.64}−0.84⋅21=-17.64.Then, e−17.64≈2.29⋅10−8e^{-17.64} \approx \textbf{2.29} \cdot 10^{-8}e−17.64≈2.29⋅10−8, which is a *very* small number!

STEP 9

Now, calculate the denominator: 1+161.7⋅(2.29⋅10−8)≈1+3.7⋅10−6≈11 + 161.7 \cdot (2.29 \cdot 10^{-8}) \approx 1 + \textbf{3.7} \cdot 10^{-6} \approx 11+161.7⋅(2.29⋅10−8)≈1+3.7⋅10−6≈1.

STEP 10

Finally, divide the numerator by the denominator: N(21)=97621≈9762N(21) = \frac{9762}{1} \approx \textbf{9762}N(21)=19762​≈9762.

STEP 11

After 21 days, approximately *all* **9762** people in Sickville will be infected with the Blue Moon Virus. bbb is **161.7** and the formula is N(t)=97621+161.7e−0.84tN(t) = \frac{9762}{1 + 161.7e^{-0.84t}}N(t)=1+161.7e−0.84t9762​.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Find b
2. Define the function
3. Calculate the result

Remember, the logistic growth formula is $N(t) = \frac{P}{1 + be^{-kt}}$ , where $P$ is the total population, $k$ is the growth rate, and $N(0)$ is the initial number of infected people.

We know $P = \textbf{9762}$ , $k = \textbf{0.84}$ , and $N(0) = \textbf{60}$ .
Let's plug these values into our formula at $t=0$ : $N(0) = \frac{9762}{1 + be^{-0.84 \cdot 0}}$ .

Now that we know $b = \textbf{161.7}$ , we can write the complete formula for the number of infected people after $t$ days: $N(t) = \frac{9762}{1 + 161.7e^{-0.84t}}$ .

We want to find the number of infected people after 21 days, so we plug $t = \textbf{21}$ into our formula: $N(21) = \frac{9762}{1 + 161.7e^{-0.84 \cdot 21}}$ .

First, calculate the exponent: $-0.84 \cdot 21 = \textbf{-17.64}$ .
Then, $e^{-17.64} \approx \textbf{2.29} \cdot 10^{-8}$ , which is a very small number!

Now, calculate the denominator: $1 + 161.7 \cdot (2.29 \cdot 10^{-8}) \approx 1 + \textbf{3.7} \cdot 10^{-6} \approx 1$ .

Finally, divide the numerator by the denominator: $N(21) = \frac{9762}{1} \approx \textbf{9762}$ .

After 21 days, approximately all 9762 people in Sickville will be infected with the Blue Moon Virus. $b$ is 161.7 and the formula is $N(t) = \frac{9762}{1 + 161.7e^{-0.84t}}$ .