Question

NO LINKS!!! URGENT HELP PLEASE!!!

3. A virus has infected 400 people in the town and is spreading to 25% more people each day. Write an exponential function to model this situation, then find the number of 3000 people are infected.

4. The population of a small town was 10,800 in 2002. Since then, the population has decreased at a rate of 2.5% each year. Write an exponential function to model the situation, then find when the popuation reaches half the 2002 value?

Answer 1

Explanation:

3. Let P(t) be the number of people infected by the virus at time t (in days). We can model the situation with the following exponential function:

P(t) = 400 * 1.25^t

Here, 400 represents the initial number of infected people, and 1.25 represents the growth factor, since the virus is spreading to 25% more people each day.

To find the number of people infected after t days, we can substitute t = (log(3000) - log(400)) / log(1.25) into the equation:

P(t) = 400 * 1.25^t

P(t) = 400 * 1.25^((log(3000) - log(400)) / log(1.25))

P(t) ≈ 2,343

Therefore, approximately 2,343 people are infected when the total number of infections reaches 3000.

4. Let P(t) be the population of the town at time t (in years). We can model the situation with the following exponential function:

P(t) = 10,800 * 0.975^t

Here, 10,800 represents the initial population in 2002, and 0.975 represents the decay factor, since the population is decreasing at a rate of 2.5% each year.

To find when the population reaches half the 2002 value, we can set P(t) = 5,400 and solve for t:

5,400 = 10,800 * 0.975^t

0.5 = 0.975^t

log(0.5) = t * log(0.975)

t ≈ 28.2

Therefore, the population will reach half the 2002 value in approximately 28.2 years, which corresponds to the year 2030.

Answer 2

3) 9.03 days

4) 27.38 years

Explanation:

Question 3

To model the spread of the virus over time, we can use an exponential function in the form:

`\large\boxed{P(t) = P_0(1 + r)^t}`

where:

• P(t) is the number of infected people after t days.
• P₀ is the initial number of infected people.
• r is the daily growth rate (as a decimal).
• t is the time elapsed (in days).

Given the virus has infected 400 people in the town and is spreading to 25% more people each day:

• P₀ = 400
• r = 25% = 0.25

Substitute these values into the formula to create a function for P in terms of t:

`P(t) = 400(1 + 0.25)^t`

`P(t) = 400(1.25)^t`

To find how many days it will take for 3000 people to be infected, set P(t) equal to 3000 and solve for t:

`\begin{aligned}P(t)&=3000\\\implies 400(1.25)^t&=3000\\(1.25)^t&=7.5 \\\ln (1.25)^t&=\ln(7.5)\\t \ln (1.25)&=\ln(7.5)\\t &=(\ln(7.5))/(\ln (1.25))\\t&=9.02962693...\end{aligned}`

Therefore, it will take approximately 9.03 days for the virus to infect 3000 people, assuming the daily growth rate remains constant at 25%.

Note: After 9 days, 2980 people would be infected. After 10 days, 3725 people would be infected.

`\hrulefill`

Question 4

To model the population of the town over time, we can use an exponential function in the form:

`\large\boxed{P(t) = P_0(1 - r)^t}`

where:

• P(t) is population after t days.
• P₀ is the initial population.
• r is the annual decay rate (as a decimal).
• t is the time elapsed (in days).

Given the initial population was 10,800 and the population has decreased at a rate of 2.5% each year:

• P₀ = 10,800
• r = 2.5% = 0.025

Substitute these values into the formula to create a function for P in terms of t:

`P(t) = 10800(1 -0.025)^t`

`P(t) = 10800(0.975)^t`

To find how many days it will take for the population to halve, set P(t) equal to 5400 and solve for t:

`\begin{aligned}P(t)&=5400\\\implies 10800(0.975)^t&=5400\\(0.975)^t&=0.5 \\\ln (0.975)^t&=\ln(0.5)\\t \ln (0.975)&=\ln(0.5)\\t &=(\ln(0.5))/(\ln (0.975))\\t&=27.3778512...\end{aligned}`

Therefore, it will take approximately 27.38 years for the population to reach half the 2002 value, assuming the annual decay rate remains constant at 2.5%.