Growing coffee

Task

The coffee variety Arabica yields about 750 kg of coffee beans per hectare, while Robusta yields about 1200 kg per hectare (reference). Suppose that a plantation has $a$ hectares of Arabica and $r$ hectares of Robusta.

1. Write an equation relating $a$ and $r$ if the plantation yields 1,000,000 kg of coffee.
2. On August 14, 2003, the world market price of coffee was $1.42 per kg of Arabica and $0.73 per kg of Robusta. Write an equation relating $a$ and $r$ if the plantation produces coffee worth $1,000,000.

IM Commentary

This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).

Solution

1. We see that $a$ hectares of Arabica will yield $750a$ kg of beans, and that $r$ hectares of Robusta will yield $1200r$ kg of beans. So the constraint equation is $$750a+ 1200r =1,\!000,\!000.$$
2. We know that $a$ hectares of Arabica yield $750a$ kg of beans worth $1.42/kg for a total dollar value of $1.42(750a)=1065a$. Likewise, $r$ hectares of Robusta yield $1200r$ kg of beans worth $0.73/kg for a total dollar value of $0.73(1200r)=876r$. So the equation governing the possible values of $a$ and $r$ coming from the total market value of the coffee is $$1065a+876r=1,\!000,\!000.$$