N(t) = 650/1+649 e^−0.5 t
(b) How many people will have heard the rumor after 4 hours?
After 9 hours?
(c) When will half the people have heard the rumor? t =
(d) When is the rumor spreading fastest? t =
Your problem doesn't state the original population, so you need to rephrase your homework question to give all the information.
You can use a program like Winplot to plot this function, and pick off the relevant data.
N(t) = 650 + 649e^-0.5t.
a. -0.5t = -0.5*4 = -2,
N(4) = 650 + 649 * e^-2,
= 650 + 649*0.1352,
= 650 + 87.8 = 737 people.
b. -0.5t = -0.5 * 9 = -4.5,
N(9) = 650 + 649*e^-4.5,
= 650 + 649 * 0.01111,
= 650 + 7.21 = 657 people.
To answer these questions, we need to plug the given values of time (t) into the equation for N(t) and calculate the corresponding number of people who have heard the rumor.
(a) Given: N(t) = 650 / (1 + 649e^(-0.5t))
(b) To find the number of people who have heard the rumor after 4 hours, substitute t = 4 into the equation:
N(4) = 650 / (1 + 649e^(-0.5*4))
Simplifying:
N(4) = 650 / (1 + 649e^(-2))
Calculate the value of N(4).
(c) To find when half the people have heard the rumor, we need to find the value of t that satisfies the equation N(t) = N(0) / 2, where N(0) is the number of people who have heard the rumor initially (t = 0).
Plug in N(0) = 650 into the equation:
N(t) = (650 / (1 + 649e^(-0.5t))) = 650 / 2
Solve for t.
(d) The rumor is spreading fastest when the derivative of N(t) with respect to t is at its maximum. So we need to find the value of t that maximizes dN(t)/dt.
First, find the derivative of N(t) with respect to t by taking the derivative of the expression in step (a).
Then, find the value of t that makes dN(t)/dt maximum.
To answer these questions, we need to use the given function:
N(t) = 650/(1+649e^(-0.5t))
Let's go through each question step by step:
(b) How many people will have heard the rumor after 4 hours?
To find the number of people who have heard the rumor after 4 hours, substitute t = 4 into the function N(t).
N(4) = 650/(1+649e^(-0.5*4))
= 650/(1+649e^(-2))
Now, you can calculate this value using a calculator or a math software. The result will be the number of people who have heard the rumor after 4 hours.
Similarly, for the next question:
(b) How many people will have heard the rumor after 9 hours?
Substitute t = 9 into the function N(t).
N(9) = 650/(1+649e^(-0.5*9))
= 650/(1+649e^(-4.5))
Calculate this value using a calculator or math software to find the number of people who have heard the rumor after 9 hours.
(c) When will half the people have heard the rumor? t =
To find when half the people have heard the rumor, we need to find the value of t that makes N(t) equal to half of the total number of people (which is 650).
So, set N(t) = 650/2 = 325:
325 = 650/(1+649e^(-0.5t))
Now, solve this equation for t. Multiply both sides by (1+649e^(-0.5t)) and then solve for t.
Once you find the value of t, that will answer the question of when half the people will have heard the rumor.
(d) When is the rumor spreading fastest? t =
To find when the rumor is spreading fastest, we need to find the maximum rate of change of N(t). We can do this by finding the derivative of N(t) with respect to t and solving for t when the derivative equals zero.
Differentiate N(t) with respect to t:
dN(t)/dt = -650 * (-0.5) * e^(-0.5t) / (1 + 649e^(-0.5t))^2
Set dN(t)/dt = 0 and solve for t to find when the rumor is spreading fastest.
Note: Solving the equations may require the use of a calculator or math software to obtain accurate values.