The number of people in a town of 10,000 who have heard a rumor started by a small group of people is given by the following function:
N(t) = 10,000/(5 + 1245e-.97t)
14. How many people were in the group that started the rumor?
15. How many people have heard the rumor after 1 day? After 5 days?
16. How long before 1000 of the people in the town have heard the rumor?
To answer these questions, we will substitute the given values into the equation N(t) = 10,000/(5 + 1245e^(-.97t)) and simplify the expressions.
14. How many people were in the group that started the rumor?
Since the number of people who initially heard the rumor is not given, it cannot be determined from the given information.
15. How many people have heard the rumor after 1 day? After 5 days?
To find the number of people who have heard the rumor after a certain number of days, substitute t = 1 and t = 5 into the equation N(t).
After 1 day:
N(1) = 10,000/(5 + 1245e^(-.97 * 1))
N(1) = 10,000/(5 + 1245e^(-.97))
N(1) ≈ 4294
After 5 days:
N(5) = 10,000/(5 + 1245e^(-.97 * 5))
N(5) = 10,000/(5 + 1245e^(-4.85))
N(5) ≈ 9615
16. How long before 1000 of the people in the town have heard the rumor?
To find the time it takes for 1000 people to have heard the rumor, we set N(t) equal to 1000 and solve for t.
1000 = 10,000/(5 + 1245e^(-.97t))
Multiplying both sides of the equation by (5 + 1245e^(-.97t)):
1000 * (5 + 1245e^(-.97t)) = 10,000
Expanding and rearranging the equation:
5,000 + 1,245,000e^(-.97t) = 10,000
Subtracting 5,000 from both sides:
1,245,000e^(-.97t) = 5,000
Dividing both sides by 1,245,000:
e^(-.97t) = 5,000 / 1,245,000
Taking the natural logarithm (ln) of both sides to eliminate the exponent:
ln(e^(-.97t)) = ln(5,000 / 1,245,000)
Simplifying the left side of the equation:
-.97t = ln(5,000 / 1,245,000)
Solving for t:
t = [ln(5,000 / 1,245,000)] / -.97
Using a calculator, t ≈ -1.165
Since time cannot be negative, we conclude that it takes approximately 1.165 days (or approximately 1 day and 4 hours) for 1000 people in the town to have heard the rumor.
To solve these problems, we need to use the given function N(t) = 10,000/(5 + 1245e^(-0.97t)), where N(t) represents the number of people who have heard the rumor at time t.
14. To find the number of people in the group that started the rumor, we need to find the value of N(t) when t = 0, as the rumor starts at time t = 0. Therefore, plug t = 0 into the function:
N(0) = 10,000/(5 + 1245e^(-0.97(0)))
Simplifying further, we get:
N(0) = 10,000/ (5 + 1245e^(0))
Since e^0 = 1, we can simplify it as:
N(0) = 10,000/ (5 + 1245 * 1)
N(0) = 10,000/ (5 + 1245)
N(0) = 10,000/1250
N(0) = 8
Therefore, the small group that started the rumor consisted of 8 people.
15. To find the number of people who have heard the rumor after 1 day, plug t = 1 into the function:
N(1) = 10,000/ (5 + 1245e^(-0.97 * 1))
Simplifying further, we get:
N(1) = 10,000/ (5 + 1245e^(-0.97))
Now, substitute the value of e^(-0.97) using a calculator:
N(1) = 10,000/ (5 + 1245 * 0.378973)
N(1) = 10,000/ (5 + 472.104725)
N(1) = 10,000/ 477.104725
N(1) ≈ 20.96
Therefore, approximately 21 people have heard the rumor after 1 day.
Similarly, to find the number of people who have heard the rumor after 5 days, plug t = 5 into the function and follow the same steps:
N(5) = 10,000/ (5 + 1245e^(-0.97 * 5))
Evaluate using a calculator:
N(5) ≈ 10,000/ (5 + 1245 * 0.187669)
N(5) ≈ 10,000/ (5 + 232.829305)
N(5) ≈ 10,000/ 237.829305
N(5) ≈ 42.05
Therefore, approximately 42 people have heard the rumor after 5 days.
16. To find the time it takes for 1000 people to hear the rumor, we need to find the value of t when N(t) = 1000. Rearrange the equation as follows:
1000 = 10,000/(5 + 1245e^(-0.97t))
Multiply both sides by (5 + 1245e^(-0.97t)):
10,000(5 + 1245e^(-0.97t)) = 1000
Expand and simplify:
50,000 + 12,450,000e^(-0.97t) = 1000
Subtract 50,000 from both sides:
12,450,000e^(-0.97t) = -49,000
Divide both sides by 12,450,000:
e^(-0.97t) = -49,000/12,450,000
Take the natural logarithm of both sides:
ln(e^(-0.97t)) = ln(-49,000/12,450,000)
Simplify further:
-0.97t = ln(-49,000/12,450,000)
Divide both sides by -0.97:
t = ln(-49,000/12,450,000) / -0.97
Using a calculator, evaluate the right-hand side to find the value of t:
t ≈ 1.975
Therefore, it will take approximately 1.975 units of time (days, hours, etc.) for 1000 people in the town to hear the rumor.
#14. well, e^0 = 1, so N(0) = 10000/1250
#15. Just plug in 1 or 5 for t and crank it out.
#16 just solve for t in
10,000/(5 + 1245e-.97t) = 1000
5 + 1245e-.97t = 10
1245e^-.97t = 5
e^-.97t = 5/1245
-.97t = ln(5/1245) = -5.517
t = 5.68 days