Math Question That I Need Help On

Question

A rumor about canceling spring break is being spread around a school. The following equation models the number of students in the school who have heard the rumor after t days:

Equation: N (t) = 1800/1+149e^-0.625t

A.). How many people started the rumor?

B. ). How many students are in the school?

C.). To the nearest 1/10, how many days will it be before the rumor spreads to ½ the student body?

D.). The following equation represents the instantaneous rate of change in students per day of the spread of the rumor. 

Another Equation: N^1 (t)=0.625N (1- N/1800)

N represents the number of students who have heard the rumor and t is time in days. 

How many students have heard the rumor when the rumor is spreading the fastest? 

What is the rate of change in students per day at this time ?

Answer 1

A) At the start of the rumor, time (t)= 0 days;

Therefore the number of people who started the rumor is:

N(0)=1800/(1+149e^-0.625*0)=1800/(1+149e⁰)=1800/(1+149*1)=1800/150=12 (Ans.)

B) To find the total number of people in the school, we need to find the value of N(t) at t tends to ∞

lim_t→∞ N(t) = lim_t→∞ {1800/(1+149e^-0.625*t)} = 1800/(1+149*0)

[This is because when t is very large, -0.625*t will tend to -∞ which suggests that e^–0.625t will tend to 0]

Therefore, lim_t→∞ N(t) = 1800

Hence, there are 1800 people in the school

C) Let's say at time t=t_1/2 half the people in the school hear the rumor.

Therefore, N(t_1/2)=900;

⇒ 1800/(1+149e^-0.625*t_1/2) = 900

⇒ 1+149e^-0.625*t_1/2 = 1800/900

⇒ 149e^-0.625*t_1/2 = 2-1

⇒ e^-0.625*t_1/2 = 1/149

Taking natural log,

-0.625*t_1/2*(ln e) = ln (1/149)

⇒ t_1/2 = -5/-0.625

Therefore, The time taken for the rumor to spread to half the school is 8 days

D) To find the maximum rate of spread, we need to find at what value of N the given equation achieves the maximum. We can do that by checking when the differentiation w.r.t N is 0.

dN^'(t)/dN = d/dN of {0.625N (1- N/1800)}

We know that for h(x)=f(x)g(x), the differentiation of h(x) is h'(x)=f'(x)g(x) + f(x)g'(x)

Therefore N"(t)=0.625(1-N/1800) + 0.625N(-1/1800) = 0.625 - 0.625N/1800 - 0.625N/1800 = 0.625 - 1.25N/1800

Therefore when N"(t) = 0

N= 0.625*1800/1.25 = 900

At N=900, N'(t) should be a maximum if N"'(t)<0. Let's see if that is true

N"'(t) = dN"(t)/dN = -1.25/1800

Hence, at N=900, N'(t) is maximum

Therefore, when the rumor is spreading the fastest, the number of students who know the rumor is 900

The rate of change per day at this time is N'(t)=0.625*900*(1-900/1800) = 562.5*0.5 = 281.25 (Ans.)