Question

Consuelo is buying flavored coffee and plain coffee. The total amount of money she spends on coffee is T=5.50p+7f,
where p represents the cost of a package of plain coffee and f represents the cost of a package of flavored coffee.

a. If Consuelo has $40 to spend on coffee, describe the constraints on the formula.

b. Solve for f.

Answer 1

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:

1. **Total Budget Constraint**:
- The total cost cannot exceed the amount she has:
\[
5.50p + 7f \leq 40
\]

2. **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 \( f \):

1. Subtract \( 5.50p \) from both sides:
\[
7f \leq 40 - 5.50p
\]

2. 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 \( f \geq 0 \).