n(t) = 800/ (1 + 39e^-t)
Where n is the number of students who have heard the rumour as a function of time, t, in days.
What is the student population of the school?
How long will it take for the rumour to reacglh half of the school's population?
What dat was the rumour spreading the fastest?
To determine the student population of the school, we need to find the value of n when time, t, is infinity.
1. Let's substitute t with infinity in the given function to find the student population of the school:
n(t) = 800 / (1 + 39e^(-t))
n(infinity) = 800 / (1 + 39e^(-infinity))
As e^(-infinity) approaches 0, the denominator approaches 1, making the student population:
n(infinity) = 800 / 1
n(infinity) = 800
So, the student population of the school is 800.
2. To determine how long it will take for the rumor to reach half of the school's population, we need to find the value of t when n(t) is equal to half of the student population.
n(t) = 800 / (1 + 39e^(-t))
n(t) = 400
800 / (1 + 39e^(-t)) = 400
Divide both sides of the equation by 400:
(1 + 39e^(-t)) = 2
By subtracting 1 from both sides of the equation, we get:
39e^(-t) = 1
To isolate e^(-t), we can divide both sides of the equation by 39:
e^(-t) = 1/39
To remove the natural logarithm, we can take the natural logarithm (ln) of both sides:
-t = ln(1/39)
Finally, we can solve for t by multiplying both sides by -1:
t = -ln(1/39)
Calculating the value of t gives us:
t ≈ 3.6639 days
So, it will take approximately 3.6639 days for the rumor to reach half of the school's population.
3. The rate at which the rumor is spreading can be determined by finding the maximum derivative of the function n(t). This can be done by differentiating the function with respect to t and finding the time at which the derivative is maximum.
n(t) = 800 / (1 + 39e^(-t))
To find the time when the rumor is spreading fastest, we need to find the maximum point of the derivative of n(t). So, let's differentiate the function n(t) with respect to t:
n'(t) = 800 * [e^(-t)] * [39e^(-t)] / [(1 + 39e^(-t))^2]
To find the critical points of this function, we need to solve for t when n'(t) equals 0:
800 * [e^(-t)] * [39e^(-t)] / [(1 + 39e^(-t))^2] = 0
Since e^(-t) cannot be zero, this equation will only be satisfied when the numerator equals zero:
800 * e^(-t) * [39e^(-t)] = 0
Simplifying the equation, we have:
39e^(-2t) = 0
This equation has no solutions. Hence, there are no critical points for t. This means there is no specific day when the rumor is spreading fastest; it is spreading at a constant rate throughout the days.
To find the student population of the school, we need to evaluate the limit of the function as t tends to infinity. Let's calculate:
lim (t→∞) n(t) = lim (t→∞) 800 / (1 + 39e^(-t))
As t approaches infinity, the exponential term in the denominator becomes negligible compared to 1. Therefore, we have:
lim (t→∞) n(t) = 800 / 1 = 800
So, the student population of the school is 800.
To determine how long it will take for the rumor to reach half of the school's population, we need to solve the equation n(t) = 400. Let's proceed:
n(t) = 800 / (1 + 39e^(-t))
Setting n(t) = 400:
400 = 800 / (1 + 39e^(-t))
Now, we can solve for t:
1 + 39e^(-t) = 800 / 400 = 2
39e^(-t) = 2 - 1 = 1
e^(-t) = 1/39
Taking the natural logarithm of both sides to eliminate the exponential:
ln(e^(-t)) = ln(1/39)
-t = ln(1/39)
t = -ln(1/39)
The value of t will be positive since it represents time. Therefore:
t ≈ 3.663 days
So, it will take approximately 3.663 days for the rumor to reach half of the school's population.
To determine the day when the rumor was spreading the fastest, we need to find the maximum rate of change of n(t) with respect to t. We can achieve this by finding the derivative of n(t) and setting it equal to zero. Let's proceed:
n(t) = 800 / (1 + 39e^(-t))
Taking the derivative with respect to t:
n'(t) = -31200e^(-t) / (1 + 39e^(-t))^2
Setting n'(t) = 0:
-31200e^(-t) / (1 + 39e^(-t))^2 = 0
Since the numerator cannot be zero, we need to solve:
e^(-t) = 0
This equation can never be satisfied since e^(-t) is always positive (as it represents an exponential function). Therefore, there is no specific day when the rumor was spreading the fastest.
Note: The derivative of n(t) represents the rate of change of the number of students who have heard the rumor at a given time. However, in this case, the derivative is always negative (n'(t) < 0), indicating a continuous decline in the number of students who have heard the rumor over time.