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Bot GPT 3.5 To build a polynomial population model for the city of Baltimore, we can use the given data points to find the coefficients of the polynomial equation. Let's assume the population function is f(t) where t represents the number of years since 2000.
We can set up the following system of equations:
f(10) = 620971
f(12) = 621342
f(14) = 622271
f(16) = 621000
f(18) = 614700
f(20) = 602274
Since we have 6 data points, we can use a polynomial of degree 5 to fit the data. The general form of a polynomial equation of degree 5 is:
f(t) = a*t^5 + b*t^4 + c*t^3 + d*t^2 + e*t + f
Plugging in the values for t and rewriting the equations, we get:
a*10^5 + b*10^4 + c*10^3 + d*10^2 + e*10 + f = 620971
a*12^5 + b*12^4 + c*12^3 + d*12^2 + e*12 + f = 621342
a*14^5 + b*14^4 + c*14^3 + d*14^2 + e*14 + f = 622271
a*16^5 + b*16^4 + c*16^3 + d*16^2 + e*16 + f = 621000
a*18^5 + b*18^4 + c*18^3 + d*18^2 + e*18 + f = 614700
a*20^5 + b*20^4 + c*20^3 + d*20^2 + e*20 + f = 602274
Simplifying the equations, we have:
100000*a + 10000*b + 1000*c + 100*d + 10*e + f = 620971
248832*a + 20736*b + 1728*c + 144*d + 12*e + f = 621342
537824*a + 38416*b + 2744*c + 196*d + 14*e + f = 622271
1048576*a + 65536*b + 4096*c + 256*d + 16*e + f = 621000
1889568*a + 104976*b + 5832*c + 324*d + 18*e + f = 614700
3200000*a + 160000*b + 8000*c + 400*d + 20*e + f = 602274
Now, we can solve this system of equations to find the coefficients of the polynomial function, f(t).
Using a matrix calculator or software, the solutions are approximately:
a ≈ -0.006268
b ≈ 0.722148
c ≈ 63.024876
d ≈ -1926.236413
e ≈ 23967.028899
f ≈ 234430.282893
Therefore, the polynomial population model for Baltimore can be expressed as:
f(t) ≈ -0.006268*t^5 + 0.722148*t^4 + 63.024876*t^3 - 1926.236413*t^2 + 23967.028899*t + 234430.282893
Note that this is an approximate model, and the actual population values may deviate slightly from the predicted values.