The population p of a small community on the outskirts of a city grows rapidly over a 20-year period: t05101520p1002004509502000

Answer:

After five years beyond the initial 20-year stretch, the population of this small community will be 4268.

Step-by-step explanation:

t | 0 | 5 | 10 | 15 | 20

p | 100 | 200 | 450 | 950 | 2000

The representation of the exponential function is:

p = aeᵏᵗ

where a and k represent constants.

Taking the natural logarithm on both sides:

In p = In aeᵏᵗ

In p = In a + In eᵏᵗ

In p = In a + kt

In p = kt + In a.

We can apply linear regression to model the relationship between In p and t, thereby determining the values of k and In a.

t | 0 | 5 | 10 | 15 | 20

p | 100 | 200 | 450 | 950 | 2000

In p | 4.605 | 5.298 | 6.109 | 6.856 | 7.601

In p = kt + In a.

y = mx + b

Here m corresponds to k and b is In a

By conducting a linear regression on the transformed linear relationship among In p and t and creating a graph of the variables, we derive the regression equation:

y = 0.151x + 4.584

The first image illustrates the equations needed for estimating the parameters of linear regression.

The second image displays the regression calculations alongside the graph of In p versus t.

By comparing

y = 0.151x + 4.584

to

In p = kt + In a.

y = In p

k = 0.151

x = t

In a = 4.584

a = 97.905

Thus, the exponential relationship between p and t is formulated as:

p = 97.905 e⁰•¹⁵¹ᵗ

In order to forecast the population for 5 years ahead from the 20-year mark, we need to find p at t=25 years.

0.151 × 25 = 3.775

p(t=25) = 97.905 e³•⁷⁷⁵ = 4268.41, which rounds down to 4268.

Hope this assists!!!