Question

50 POINTS!!!!!! PLEASE ANSWER THESE 2 QUESTIONS

Answer 1

(a) The equation that estimates the population of the town t years after 2010 is P(t) = 5000 * e^(0.014t), and (b) the estimated population of the town in 2022 is approximately 5935.

Explanation:

(a) The exponential growth model is given by the formula:

P(t) = P0 * e^(rt)

where:

P(t) is the future population,

P0 is the initial population,

r is the growth rate (in decimal form),

t is the time (in years),

and e is the base of the natural logarithm, approximately equal to 2.71828.

In this case, the initial population P0 is 5000, the growth rate r is 1.4% or 0.014 (in decimal form), and t is the number of years after 2010.

So, the equation that estimates the population t years after 2010 is:

P(t) = 5000 * e^(0.014t)

(b) To estimate the population of the town in 2022, we need to find the value of P(t) when t = 2022 - 2010 = 12 years.

Substituting t = 12 into the equation:

P(12) = 5000 * e^(0.014 * 12)

= 5000 * e^0.168

= 5000 * 1.183

= 5935

So, the estimated population of the town in 2022 is approximately 5935.

Answer 2

a.
`P(x) = 5000(1.014)^(t)`

b.
`P(12) = 5907`

Explanation:

Given initial population = 5000

rate of increase (common ratio) = (100% + 1.4%) or 1.014

We can use geometric formula to build an equation:

Geometric formula:
`\sf a_n = a r^(n-1)` where a is initial value, r is common ratio, n is the term position. Here the term position is years after 2010.

Equation:

`P(x) = 5000(1.014)^(2010+t-2010)`

`P(x) = 5000(1.014)^(t)`

Population in 2022. (2022-2010) = 12 years. the value of t = 12.

`P(12) = 5000(1.014)^(12) = 5907.8 = 5907`