The table gives the population of the United States, in millions, for the years

Question

The table gives the population of the United States, in millions, for the years 1900-2000.

(a) Use the exponential model and the census figures for 1900 and 1910 to predict the population in 2000. (Round your answer to the nearest million.)
P(2000) = _______ million

(b) Use the exponential model and the census figures for 1960 and 1970 to predict the population in 2000. Compare with the actual population. Then use this model to predict the population in the years 2010 and 2020. (Round your answers to the nearest million.)

Year Population Year Population 1900
1910
1920
1930
1940
1950 76
92
106
123
131
150 1960
1970
1980
1990
2000
179
203
227
250
275

Explanation / Answer

The general form of the exponential function is :

P = Ae^(kt) ------> (1)

Where, A = Initial value at time t = 0
       k = Exponential growth/decay constant
       t = time in time units
       P = Value at time t in time units

a>
We need to find the exponential model using 1900 as the start time in years that is t = 0

and the 1910 will be equivalent to , t = 10 years

Now from the table we have

In the year 1900 , that is P(0) = A = 76 ----> (2)

and in the year 1910 , that is P(10) = 92

and t = 10

Now, plug these values in equation (1)

=> 92 = 76e^(10k)

23/19 = e^(10k)

take log both sides

=> ln(23/19) = 10kln(e)

or ln(23/19) = 10k

=> k = 1/10 * ln(23/19) = 0.0191 (approximate value)   -----> (3)

From (1) , (2) and (3)

P = 76e^(0.0191t)   , -----> This is the required exponential model

Now using this model we need to find the population for the year 2000 , that is t = 100

PLug t = 100 in P = 76e^(0.0191t)

=> P = 76e^(0.0191*100) = 513.23 = 513 (Answer in nearest million)

Hence from population prediction for the year 2000 using this model is = 513 million

We need to find the exponential model using 1960 as the start time in years that is t = 0

and the year 1970 will be equivalent to , t = 10 years

Now from the table we have

In the year 1960 , that is P(0) = A = 179 ----> (2)

and in the year 1970 , that is P(10) = 203

and t = 10

Now, plug these values in equation (1)

=> 203 = 179e^(10k)

203/179 = e^(10k)

take log both sides

=> ln(203/179) = 10kln(e)

or ln(203/179) = 10k

=> k = 1/10 * ln(203/179) = 0.0126 (approximate value)   -----> (3)

From (1) , (2) and (3)

P = 179e^(0.0126t)   , -----> This is the required exponential model

Now using this model we need to find the population for the year 2000 , that is t = 40

PLug t = 40 in P = 179e^(0.0126t)

=> P(2000) = P(40) = 179e^(0.0126*40) = 296.30 = 296 (Answer in nearest million)

Hence from population prediction for the year 2000 using this model is = 296 million.

Now the actual population as per the table is 275 million.

The difference in population is = Actual - Prediction = 275 - 296 = -21 million.

The negative sign shows that the predicted population using this model is on the higher
side of the actual value.

NOw we have, P = 179e^(0.0126t)

The population in the year 2010 that is , t = 2010 - 1960 = 50 years is :

P(2010) = P(50) = 179e^(0.0126*50) = 336.01 = 336 million (approximate value)

The population in the year 2020 that is , t = 2020 - 1960 = 60 years is :

P(2020) = P(60) = 179e^(0.0126*60) = 381.22 = 381 million (approximate value)

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