To tackle this problem, let's analyze the situation step by step.
Part a: Constraints on the Formula
Consuelo's total expenditure on coffee can be represented by the formula:
\[ T = 5.50p + 7f \]
where:
- \( p \) is the number of packages of plain coffee,
- \( f \) is the number of packages of flavored coffee,
- \( T \) is the total amount she can spend, which is $40.
The constraints for this equation would include:
-
Total Budget Constraint:
- The total cost cannot exceed the amount she has: \[ 5.50p + 7f \leq 40 \]
-
Non-negativity Constraints:
- She cannot purchase a negative number of packages of coffee, hence: \[ p \geq 0 \] \[ f \geq 0 \]
Putting it together, the constraints can be described as: \[ 5.50p + 7f \leq 40, \quad p \geq 0, \quad f \geq 0 \]
Part b: Solve for \( f \)
To solve for \( f \) in terms of \( p \) using the constraint equation, we start from:
\[ 5.50p + 7f \leq 40 \]
Isolating \( f \):
-
Subtract \( 5.50p \) from both sides: \[ 7f \leq 40 - 5.50p \]
-
Divide both sides by 7: \[ f \leq \frac{40 - 5.50p}{7} \]
Thus, the solution for \( f \) is: \[ f \leq \frac{40 - 5.50p}{7} \]
This expresses \( f \) in terms of \( p \) with the understanding that it must also satisfy the non-negativity constraint \( f \geq 0 \).