Question

16

I
The latest rare disease, Expy, has entered your math classroom of 30. The students
affected by the disease is modeled by the logistic function P(t) =
150
5+25e 0.45t
where P(t) represents the number of students that understand exponential and t is time in
days.
Clearly label each answer part. Show your initial set-up and give your final answer.
Each step does not need to be shown.
A) How many students understand exponential functions initially? (6 points)
B) How many days will it take for 10 students to understand exponential functions? (6
points, multiple-choice)
f) t=(In5/2)/0.45
g) t=(In10)/0.45
h) t=0.45/(In5/2)
j) t=0.45/(In10)

Answer 1

A) Initially, 5 students understand exponential functions.

B) It will take approximately 3.04 days for 10 students to understand exponential functions.

Step-by-step explanation:

A) To find the initial number of students who understand exponential functions, we can substitute t = 0 into the equation P(t) = {150}/{5+25e^{0.45t}}.

Plugging in t = 0, we get P(0) = {150}/{5+25e^0}

= {150}/{30} = 5.

Therefore, initially, 5 students understand exponential functions.

B) To find the number of days it will take for 10 students to understand exponential functions, we need to solve the equation P(t) = 10.

Substituting P(t) = 10 and solving for t gives us:

10 = {150}/{5+25e^{0.45t}}

Multiplying both sides by 5+25e^{0.45t}:

10(5+25e^{0.45t}) = 150

Dividing both sides by 10:

5+25e^{0.45t} = 15

Subtracting 5 from both sides:

25e^{0.45t} = 10

Dividing both sides by 25:

e^{0.45t} = {10}/{25} = {2}/{5}

Taking the natural logarithm of both sides:

0.45t = ln{2/}{5}

Dividing both sides by 0.45:

t = {ln{2}/{5})} / {0.45}

Using a calculator, we can find that t is approximately 3.04 days.