F(T)=. exponential or linear? predict and find half life of #

Question

The numbers of polio cases in the world are shown in the table for various years.

Year Number of polio cases (thousands)

1988 350

1992 138

1996 33

2000 4

2005 3.2

2007 1.3

Let f(t) be the number of polio cases in thousands in the year t since 1980

Use a graphing calculator to draw a scattergram of the data. Is it better to model the data by using a linear or exponential model?

F(T)= ____ (round numbers to 2 decimal places)

predict the number of polio cases in 2014

find the approximate half life of the number of polio cases

Answer 1

It's exponential. To be linear, it would have to go down by about the same amount every four years.

In the first four years, it decreases by 212.

In the next four years, it decreases by 105

In the next four years, it decreases by 29.

That's not linear.

This video explains how to use the TI 84 type calculator to find this: https://www.youtube.com/watch?v=QfBvEjBz_1s&ab_channel=Mathispower4u

I used an online regression calculator: https://stats.blue/Stats_Suite/exponential_regression_calculator.html

F(T) = 342.58(.74^x) with x the number of years since 1988

The calculator shows the correlation (r) = -.98 and r^2 = .96. That's close to 1, so it is a strong fit for this exponential model.

2014 is 26 years since 1988: 342.58(.74^26) = .14 * 1000 = about 140 cases. (Remember, the numbers shown are in thousands, so multiply by 1000.)

Half life means it is 1/2 of the start.

342.58/2 = 171.29

171.29 = 342.58(.74^x)

.5 = .74^x

ln(.5) = xln(.74)

ln(.5)ln(.74) = x = about 2.30 years