Answer:
a) 2.303
b)
c) 1.3609
e)
f) 4.5 thousand people(smaller)
7.5 thousand people (larger)
e) t=7.93 days (smaller)
t=10.06 days (larger)
g) 1.44
Step-by-step explanation:
The given function is
To find how many thousand people have heard the news after 6 days, we substitute to get:
thousands.
b) We rewrite to get:
We differentiate using the chain rule to obtain:
This simplifies to:
We rewrite as positive index to get:
c) To find the rate at which the news is spreading after 8 days, we substitute t=8 into to get:
d) The solution to the logistic differential equation:
is
By comparison, M=12, and
The differential equation is:
e) To how many people have heard the news when its rate of spread is 1.35 thousand per day, we equate the differential equation to 1.35 and so for t first.
This gives us: t=10.06 ot t=7.93
We substitute the times into the function to get:
=4.5 thousand: smaller value
thousand: Larger value
f) The two times that the news is spreading at rate 1.35 thousand people per day are:
days: Smaller value
: Larger value
g) To find the fastest rate at which the news spread we take the second derivative and equate it to zero.
This corresponds to where the horizontal line is tangent to
From the graph the point of tangency is:
(8.99,1.44)
Therefore the fastest rate at which the news spread is 1.44
