Math  /  Calculus

QuestionFor the following exercises, use this scenario: The equation N(t)=4801+49e0.55tN(t)=\frac{480}{1+49 e^{-0.55 t}} models the number of people in a town who have heard a rumor after tt days.
How many people started the rumor? \square To the nearest whole number, how many people will have heard the rumor after 3 days? \square As tt increases without bound, what value does N(t)N(t) approach? Interpret your answer. \square This is Select an answer

Studdy Solution

STEP 1

What is this asking? We're given a formula that tells us how many people have heard a rumor after a certain number of days, and we want to figure out how many people started the rumor, how many people will have heard it after 3 days, and what happens in the long run. Watch out! Make sure to correctly interpret the meaning of tt in the formula and be careful with rounding!

STEP 2

1. Initial Rumor Spreaders
2. Rumor Spread after 3 Days
3. Long-Term Rumor Reach

STEP 3

To find out how many people started the rumor, we need to look at what happens when t=0t = 0 because tt represents the number of days since the rumor started.

STEP 4

**Substitute** t=0t = 0 into our equation: N(0)=4801+49e0.550N(0) = \frac{480}{1 + 49e^{-0.55 \cdot 0}}

STEP 5

Since any number raised to the power of zero is one, e0.550=e0=1e^{-0.55 \cdot 0} = e^0 = 1.
So, our equation becomes: N(0)=4801+491N(0) = \frac{480}{1 + 49 \cdot 1}

STEP 6

**Simplify** the denominator: N(0)=4801+49=48050N(0) = \frac{480}{1 + 49} = \frac{480}{50}

STEP 7

**Calculate the result**: N(0)=48050=485=9.6N(0) = \frac{480}{50} = \frac{48}{5} = 9.6

STEP 8

Since we're talking about people, we need a whole number.
It makes sense to round down to the nearest whole number, since you can't have a fraction of a person.
So, approximately **10** people started the rumor.

STEP 9

Now, we want to find out how many people will have heard the rumor after 3 days.
This means we need to **substitute** t=3t = 3 into our equation: N(3)=4801+49e0.553N(3) = \frac{480}{1 + 49e^{-0.55 \cdot 3}}

STEP 10

**Simplify** the exponent: N(3)=4801+49e1.65N(3) = \frac{480}{1 + 49e^{-1.65}}

STEP 11

**Calculate** the value of e1.65e^{-1.65}: e1.650.192e^{-1.65} \approx 0.192

STEP 12

**Substitute** this value back into our equation: N(3)4801+490.192N(3) \approx \frac{480}{1 + 49 \cdot 0.192}

STEP 13

**Simplify** the denominator: N(3)4801+9.40848010.408N(3) \approx \frac{480}{1 + 9.408} \approx \frac{480}{10.408}

STEP 14

**Calculate the result**: N(3)46.12N(3) \approx 46.12 Rounding to the nearest whole number, approximately **46** people will have heard the rumor after 3 days.

STEP 15

To figure out what happens to N(t)N(t) as tt gets very large, we need to think about what happens to e0.55te^{-0.55t} as tt approaches infinity.

STEP 16

As tt gets larger and larger, 0.55t-0.55t becomes a very large negative number.
Since ee is raised to a large negative number, e0.55te^{-0.55t} approaches zero.

STEP 17

So, as tt approaches infinity, our equation becomes: N(t)=4801+490N(t) = \frac{480}{1 + 49 \cdot 0}

STEP 18

**Simplify** the denominator: N(t)=4801+0=4801N(t) = \frac{480}{1 + 0} = \frac{480}{1}

STEP 19

**Calculate the result**: N(t)=480N(t) = 480 This means that as time goes on, the number of people who hear the rumor approaches **480**.
This is the maximum number of people who will eventually hear the rumor.

STEP 20

Initially, approximately **10** people started the rumor.
After 3 days, about **46** people will have heard it.
In the long run, the number of people who hear the rumor will approach **480**.

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