Let $x$ be the number of Costless vitamins Jessica buys, and $y$ be the number of Savemore vitamins she buys. We want to minimize the cost of vitamins while meeting the vitamin requirements.

Step 1: Constraints

We are given the following constraints for the amounts of vitamins A, B, and C that Jessica needs:

• Vitamin A: $3x + y \geq 60$
• Vitamin B: $x + y \geq 40$
• Vitamin C: $2x + 5y \geq 140$

Step 2: Objective function

The cost of the vitamins is:

• Costless vitamin: 5 cents each
• Savemore vitamin: 7 cents each

So, the objective function to minimize is: $\text{Cost} = 5x + 7y$

Step 3: Solving the system

We now solve this system of inequalities using methods such as graphing or linear programming, but first let's transform the inequalities into equalities to find possible boundary points.

1. $3x + y = 60$
2. $x + y = 40$
3. $2x + 5y = 140$

Solving two equations at a time:

• From equation 2, $y = 40 - x$, substitute into equation 1: $3x + (40 - x) = 60$ Simplify: $3x - x + 40 = 60$ $2x = 20 \implies x = 10$ Substituting $x = 10$ into $y = 40 - x$, we get $y = 30$.

• Now substitute $x = 10$ and $y = 30$ into equation 3: $2(10) + 5(30) = 20 + 150 = 170$ But the requirement is $140$, so this is valid.

Thus, $x = 10$ and $y = 30$ satisfies the constraints. Now, let's compute the cost: $\text{Cost} = 5(10) + 7(30) = 50 + 210 = 260 \text{ cents}$ $\text{Cost in dollars} = \frac{260}{100} = 2.60$

Final Answer:

The minimum cost for Jessica to purchase all the vitamins she needs is $2.60.

Would you like a breakdown of the method, or do you have any questions?

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Here are 5 related questions:

1. How would the cost change if Jessica's vitamin requirements increased?
2. What if the price of the vitamins changes, how would that impact the solution?
3. How would you solve this problem using graphical methods?
4. Can linear programming methods like simplex be applied here?
5. How would introducing a new brand of vitamins affect the optimization?

Tip: In optimization problems, identifying the constraints and objective function clearly is crucial to finding the minimum or maximum solution effectively.