The population p of a small community on the outskirts of a city grows rapidly over a 20-year period: t05101520p1002004509502000
As an engineer working for a utility company, you must forecast the population 5 years into the future in order to anticipate the demand for power. Employ an exponential model and linear regression to make this prediction. (Round the final answer to the nearest whole number.)
1 answer:
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!!!
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