Short simple statement about their everyday life.
Data downloaded on 2/19/2020 from hitps://ourworldindata.org/grapher/iteracy-rate-adults?tab=chart&time=1973..2016.
| Year | 1976 | 2001 | 2012 |
| ---- | --------- | --------- | --------- |
| Literacy Rate | $63.2 \%$ | $86.7 \%$ | $94.5 \%$ |
When answering the questions below, round to four decimal places in your intermediate computations.
Use interpolation or extrapolation (whichever is appropriate) to predict the literacy rate in Bolivia in 1992. Round your answer to one decimal place.
Use interpolation or extrapolation (whichever is appropriate) to predict the literacy rate in Bolivia in 2050. Round your answer to one decimal place.
Is your 2050 prediction realistic? You must select all correct responses at the same time to receive credit for this problem.
A. Yes, the 2050 prediction is realistic because it was made using a mathematical formula.
B. Yes, the 2050 prediction is realistic because it is possible for a literacy rate to exceed $100 \%$.
C. No, the 2050 prediction is not realistic because the literacy rate cannot exceed $100 \%$.
D. No, the 2050 prediction is not realistic because the literacy rate will not grow at the exact same rate for more than 40 years.
Answer (1)
Use linear interpolation to predict the literacy rate in 1992: y = 63.2 + ( 1992 − 1976 ) × ( 2001 − 1976 ) ( 86.7 − 63.2 ) = 78.24 .
Use linear extrapolation to predict the literacy rate in 2050: y = 86.7 + ( 2050 − 2001 ) × ( 2012 − 2001 ) ( 94.5 − 86.7 ) = 121.445 .
Assess the realism of the 2050 prediction: The prediction is not realistic because the literacy rate cannot exceed 100% and the growth rate is unlikely to remain constant.
State the final answers: The literacy rate in 1992 is approximately 78.2% , and the 2050 prediction is not realistic.
Explanation
Problem Analysis We are given the literacy rates in Bolivia for the years 1976, 2001, and 2012. We need to predict the literacy rate for 1992 using interpolation and for 2050 using extrapolation. Finally, we need to assess the realism of the 2050 prediction.
Interpolation Formula To predict the literacy rate in 1992, we use linear interpolation between the years 1976 and 2001. The formula for linear interpolation is: y = y 1 + ( x − x 1 ) × ( x 2 − x 1 ) ( y 2 − y 1 ) where: x = year to predict (1992) x 1 = year 1 (1976) x 2 = year 2 (2001) y 1 = literacy rate in year 1 (63.2%) y 2 = literacy rate in year 2 (86.7%)
Calculating Interpolated Value Plugging in the values, we get: y = 63.2 + ( 1992 − 1976 ) × ( 2001 − 1976 ) ( 86.7 − 63.2 ) y = 63.2 + ( 16 ) × ( 25 ) ( 23.5 ) y = 63.2 + 16 × 0.94 y = 63.2 + 15.04 y = 78.24 So, the predicted literacy rate in 1992 is 78.24%.
Extrapolation Formula To predict the literacy rate in 2050, we use linear extrapolation based on the literacy rates in 2001 and 2012. The formula for linear extrapolation is the same as interpolation: y = y 1 + ( x − x 1 ) × ( x 2 − x 1 ) ( y 2 − y 1 ) where: x = year to predict (2050) x 1 = year 1 (2001) x 2 = year 2 (2012) y 1 = literacy rate in year 1 (86.7%) y 2 = literacy rate in year 2 (94.5%)
Calculating Extrapolated Value Plugging in the values, we get: y = 86.7 + ( 2050 − 2001 ) × ( 2012 − 2001 ) ( 94.5 − 86.7 ) y = 86.7 + ( 49 ) × ( 11 ) ( 7.8 ) y = 86.7 + 49 × 0.709090909 y = 86.7 + 34.7454545 y = 121.4454545 So, the predicted literacy rate in 2050 is approximately 121.4%.
Realism Assessment The 2050 prediction is not realistic because the literacy rate cannot exceed 100%. Also, it is unlikely that the literacy rate will grow at the exact same rate for more than 40 years.
Final Answer The predicted literacy rate in Bolivia in 1992 is 78.2%. The predicted literacy rate in Bolivia in 2050 is 121.4%. The 2050 prediction is not realistic because literacy rate cannot exceed 100% and the growth rate is unlikely to remain constant.
Examples
Linear interpolation and extrapolation are used in various fields such as finance, statistics, and engineering to estimate values based on existing data. For example, in finance, interpolation can be used to estimate the yield of a bond with a maturity date that falls between two bonds with known yields. In environmental science, extrapolation can be used to predict future climate conditions based on historical data. These techniques help in making informed decisions and predictions when complete data is not available.
