Question 3.5.7: Comparing Linear and Exponential Models The data for world p...
Comparing Linear and Exponential Models
The data for world population are shown in Table 3.10. Using a graphing utility’s linear regression feature and exponential regression feature, we enter the data and obtain the models shown in Figure 3.28.
Because r, the correlation coefficient, is close to 1 in each screen in Figure 3.28, the models fit the data very well.
a. Use Figure 3.28 to express each model in function notation, with numbers rounded to three decimal places.
b. How well do the functions model world population in 2000?
c. By one projection, world population is expected to reach 8 billion in the year 2026. Which function serves as a better model for this prediction?
Table 3.10
| Although the domain of y = ab^x is the set of all real numbers, some graphing utilities only accept positive values for x. That’s why we assigned x to represent the number of years after 1949. | x, Number of Years after 1949 |
y, World Population (billions) |
| 1 (1950) | 2.6 | |
| 11 (1960) | 3.0 | |
| 21 (1970) | 3.7 | |
| 31 (1980) | 4.5 | |
| 41 (1990) | 5.3 | |
| 51 (2000) | 6.1 | |
| 61 (2010) | 6.9 |
a. Using Figure 3.28 and rounding to three decimal places, the functions
f (x) = 0.074x + 2.294 and g(x) = 2.577(1.017)^x
model world population, in billions, x years after 1949. We named the linear function f and the exponential function g, although any letters can be used.
b. Table 3.10 on the previous page shows that world population in 2000 was 6.1 billion. The year 2000 is 51 years after 1949. Thus, we substitute 51 for x in each function’s equation and then evaluate the resulting expressions with a calculator to see how well the functions describe world population in 2000.
f (x) = 0.074x + 2.294 This is the linear model.
f (51) = 0.074(51) + 2.294 Substitute 51 for x.
≈ 6.1 Use a calculator.
g(x) = 2.577(1.017)^x This is the exponential model.
g(51) = 2.577(1.017)^{51} Substitute 51 for x.
≈ 6.1 Use a calculator: 2.577 × 1.017 y^x (or ∧ ) 51 =.
Because 6.1 billion was the actual world population in 2000, both functions model world population in 2000 extremely well.
c. Let’s see which model comes closer to projecting a world population of 8 billion in the year 2026. Because 2026 is 77 years after 1949 (2026 – 1949 = 77), we substitute 77 for x in each function’s equation.
f (x) = 0.074x + 2.294 This is the linear model.
f (77) = 0.074(77) + 2.294 Substitute 77 for x.
≈ 8.0 Use a calculator.
g(x) = 2.577(1.017)^x This is the exponential model.
g(77) = 2.577(1.017)^{77} Substitute 77 for x.
≈ 9.4 Use a calculator: 2.577 × 1.017 y^x (or ∧ ) 77 =.
The linear function f (x) = 0.074x + 2.294 serves as a better model for a projected world population of 8 billion by 2026.