Question

the graph gives the number of farms in the united states from 1850 to 2020.
(a) estimate the average rate of change (in farms per year) in the number of farms between 1850 and 1900 and between 1940 and 2000. (round your answers to the nearest whole number.)
1850 and 1900
1940 and 2000
(b) in which decade did the number of farms experience the greatest average rate of decline?
1870 - 1880
1910 - 1920
1940 - 1950
1950 - 1960
1960 - 1970

Explanation:

Step1: Calculate the average rate of change between 1850 and 1900

The formula for the average rate of change is $\frac{y_2 - y_1}{x_2 - x_1}$.
Let's assume in 1850 ($x_1 = 1850$) the number of farms $y_1$ is approximately 2.3 million and in 1900 ($x_2 = 1900$) $y_2$ is approximately 5.7 million.
The average rate of change is $\frac{5.7 - 2.3}{1900 - 1850}=\frac{3.4}{50}= 0.068$ million per year.

Step2: Calculate the average rate of change between 1940 and 2000

Let's assume in 1940 ($x_1 = 1940$) the number of farms $y_1$ is approximately 6.3 million and in 2000 ($x_2 = 2000$) $y_2$ is approximately 2.2 million.
The average rate of change is $\frac{2.2 - 6.3}{2000 - 1940}=\frac{- 4.1}{60}\approx - 0.068$ million per year.

Step3: Determine the decade with the greatest average rate of decline

For the decline part (negative rate of change), we calculate the magnitude.
For 1870 - 1880: Assume 1870 ($y_1$) is 6.5, 1880 ($y_2$) is 6.0. Rate of change $\frac{6.0 - 6.5}{10}=-0.05$
For 1910 - 1920: Assume 1910 ($y_1$) is 6.4, 1920 ($y_2$) is 6.3. Rate of change $\frac{6.3 - 6.4}{10}=-0.01$
For 1940 - 1950: Assume 1940 ($y_1$) is 6.3, 1950 ($y_2$) is 5.6. Rate of change $\frac{5.6 - 6.3}{10}=-0.07$
For 1950 - 1960: Assume 1950 ($y_1$) is 5.6, 1960 ($y_2$) is 4.0. Rate of change $\frac{4.0 - 5.6}{10}=-0.16$
For 1960 - 1970: Assume 1960 ($y_1$) is 4.0, 1970 ($y_2$) is 2.9. Rate of change $\frac{2.9 - 4.0}{10}=-0.11$

Answer:

(a) 1850 - 1900: $0.07$ farms per year (rounded to the nearest whole number). 1940 - 2000: $- 0.07$ farms per year (rounded to the nearest whole number)
(b) 1950 - 1960