Question

select the correct answer.

the cohen family started their wheat farm in 1995. the equation models the number of bushels of wheat they produced each year, where \\(t\\) is the number of years since 1995.

\\c(t) = 7,000 + 500 \ln(t + 1)\\

the mason family also started their wheat farm in 1995. the graph models the number of bushels of wheat they produced each year, where \\(t\\) is the number of years since 1995.

which statement is true about the quantity of wheat the two farms will produce as the years pass?

• the mason farms wheat production will continue to increase each year without bound, while the cohen farms wheat production will approach a stable amount.
• the wheat production of both farms will approach a stable amount as the years pass.
• the cohen farms wheat production will continue to increase each year without bound, while the mason farms wheat production will approach a stable amount.

Explanation:

Analyze the behavior of the Cohen family's wheat production function

$$ C(t) = 7,000 + 500\ln(t + 1) $$

As \(t \to \infty\), the natural logarithm function \(\ln(t + 1) \to \infty\). Therefore, \(C(t)\) increases without bound.

Analyze the behavior of the Mason family's wheat production graph

The graph of \(M(t)\) shows a horizontal asymptote as \(t\) increases. The curve levels off and approaches a stable value near \(12,000\) bushels.

Compare the long-term behaviors to identify the correct statement

The Cohen farm's production increases without bound, while the Mason farm's production approaches a stable amount.

Answer:

• The Mason farm's wheat production will continue to increase each year without bound, while the Cohen farm's wheat production will approach a stable amount.
• The wheat production of both farms will approach a stable amount as the years pass.
• The Cohen farm's wheat production will continue to increase each year without bound, while the Mason farm's wheat production will approach a stable amount. (Correct answer)