Substituting , yields
, and
which can be separated as
Integrate both sides concerning . For the left integral, split into partial fractions:
Resolve for :
Substitute and determine
:
Utilize the given initial condition to find :
concluding with the particular solution:
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!!!
