Question

1. In 2005 a town with a population of 10,000 finds that its population is declining by

a rate of 0.3% per year. Find the following:

a. Find the exponential function for the given information.
Answer: P(t) = 10,000e ^-.003t

b. Estimate the population of the town in 2016.
In a Complete sentence.
Answer: The population in 2016 is approximately
9,970.

c. Estimate when the population will be 8,000.
In a Complete sentence.

d. Given the above graph as well as your equation, will the population ever
be zero, and if so when?
Complete Sentence.

e. I’m a short paragraph, describe the general trend of the data you have collected in the above questions and use your answers to justify the trend.

I need help with c, d, and e. Reference Attached photo

Answer 1

c. The population is estimated to reach 8,000 around year 45.37, suggesting it will occur between years 45 and 46.

d. The population will never be zero as the exponential function never reaches zero.

e. The data indicates a continuous population decline, approaching zero asymptotically over time.

c. To estimate when the population will be 8,000, set the exponential function equal to 8,000 and solve for t:

`\[10,000e^(-0.003t) = 8,000\]`

Divide both sides by 10,000:

`\[e^(-0.003t) = 0.8\]`

Take the natural logarithm of both sides:

`\[-0.003t = \ln(0.8)\]`

Solve for t:

`\[t = (\ln(0.8))/(-0.003) \approx 45.37\]`

So, the population is estimated to be 8,000 around year 45.37, which implies the population will reach this level between years 45 and 46.

d. The exponential function
`\(P(t) = 10,000e^(-0.003t)\)` never reaches zero because the exponential term never equals zero. The population will continue to decline indefinitely, but it will never completely vanish.

e. The data indicates a declining population, as reflected by the negative exponential function coefficient. The estimate for 2016 is consistent with this trend, showing a decrease from the initial population. The population will continue to decrease, never reaching zero but approaching it asymptotically. This trend aligns with the nature of exponential decay, indicating a continuous decline over time.