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Indian population The table gives the midyear population of India (in millions) for the last half of the 20th century.(a) Make a scatter plot, semilog plot, and log-log plot for these data and comment on which type of model would be most appropriate.(b) Obtain an exponential model for the population.(c) Use your model to estimate the population in 2010 and compare with the actual population of 1173 million. What conclusion can you make?

(a) See explanation for graphs; Exponential(b) $y=\left(2.2761 \cdot 10^{-15}\right)\left(1.02052861^{x}\right)$(c) 1247 million people; the model overestimates the population

Calculus 1 / AB

Chapter 1

Functions and Sequences

Section 5

Logarithms; Semilog and Log-Log Plots

Functions

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first we must make a scatter plot semi log plot and log log plot for our data. So here I've done this in days most but you can use any graphing tool available to you. This is our base data, the plane data, just the year values and the population values in millions given to us. And so it makes this nice scatter plot here in black. Now for the semi log plot we have taken the base 10 logarithms of our population values and then plotted them against the plane year values. So now we have this blue semi log plot. And finally for the log log plot we used We took the base 10 logarithms of both the years and the population values. And so it gives us this nice red log log plot. Now to find the best model for our data, we can observe each of these plots and see how well they fit to align. If the linear, if the scatter plot fits best to align, then we should best use a linear model. But that does not seem to be the case here, as there's a little bit of a curve, as you can see in the semi log plot. If the semi log plot best fits to a line and we should use an exponential model, it does look pretty good if the log log plot fits best to headline, we should use a power model, but out of the three of these, the semi log plot appears to be the most linear. So we should use an exponential model for this data. Now we must generate the exponential model for this data. And so we can also do this in dez most. So here this is the data straight from the problem. This is also the data we used for the scatter plot problem. And so you can use Dismas to generate an exponential model, like. So it will give you the coefficients here in terms of A. And B. And so you can see that a is about 2.276, 1 Times 10 to the -15. And I forgot to write exponential his vest. Sorry about that. And then our B value is about 1.02053. If we round it off here 1.02053 for the final part of our problem, we have to estimate The population in 2010 using our model. So we plug in the coefficients we found in the last part And then plug in 2010 for X as our year. And if you plug this into your calculator and round up to the nearest million, you will get about 1,274 million people. And so compared to the actual population of 1,173 million people, Our models slightly overestimates. So from this we can deduce that our population will approximately overestimate the population for a given year, right a year.

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