Problem 320 - Elementary Algebra

Question

Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs \(\$ 3.20\) and a protein bar costs \(\$ 1.75 .\) The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has \(\$ 10\) dollars to spend (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Could he buy 3 bottles of protein water and 1 protein bar? (a) Could he buy no bottles of protein water and 5 protein bars?

Step-by-Step Solution

Answer

The system of inequalities is:\begin{cases}27x + 16y \geq 80\3.20x + 1.75y \leq 10\x \geq 0\y \geq 0\d (end.Mark could not buy 3 bottles of protein water and 1 protein bar. He could buy no bottles of protein water and 5 protein bars.

Step 1: Define Variables

Let's define the variables to represent the number of bottles of protein water and protein bars Mark buys.\(x\) = number of bottles of protein water.\(y\) = number of protein bars.

Step 2: Set Up Inequalities for Protein Requirement

Mark needs to consume at least 80 grams of protein a day. Each bottle of protein water provides 27 grams and each protein bar provides 16 grams. This gives the inequality:\[27x + 16y \geq 80\]

Step 3: Set Up Inequalities for Budget

Mark has \(\$10\) to spend. Each bottle of protein water costs \(\$3.20\) and each protein bar costs \(\$1.75\). This gives the inequality:\[3.20x + 1.75y \leq 10\]

Step 4: Include Non-negativity Constraints

Mark cannot buy a negative number of items. Thus, we include the constraints:\[x \geq 0\]\[y \geq 0\]

Step 5: System of Inequalities

Combine the inequalities to form the complete system:\[\begin{cases}27x + 16y \geq 80\3.20x + 1.75y \leq 10\x \geq 0\y \geq 0\end{cases}\]

Step 6: Graph the System

Graph the inequalities on a coordinate plane.- Plot the line for \(27x + 16y = 80\) and shade the region above the line.- Plot the line for \(3.20x + 1.75y = 10\) and shade the region below the line.- Ensure the solution lies in the first quadrant where both \(x\) and \(y\) are non-negative.

Step 7: Check Specific Purchase (3 Bottles, 1 Bar)

Substitute \(x = 3\) and \(y = 1\) into the inequalities:- For protein: \(27(3) + 16(1) = 81 + 16 = 97 > 80\), so it satisfies the protein requirement.- For budget: \(3.20(3) + 1.75(1) = 9.60 + 1.75 = 11.35 > 10\), so it exceeds the budget. Therefore, he could not make this purchase.

Step 8: Check Specific Purchase (0 Bottles, 5 Bars)

Substitute \(x = 0\) and \(y = 5\) into the inequalities:- For protein: \(27(0) + 16(5) = 0 + 80 = 80\), so it exactly meets the protein requirement.- For budget: \(3.20(0) + 1.75(5) = 0 + 8.75 = 8.75 < 10\), so it is within the budget. Therefore, he could make this purchase.

Key Concepts

system of inequalities

A system of inequalities involves multiple inequalities that are considered together. In Mark's case, he must meet both his protein requirement and his budget constraint. This gives us two inequalities:

• The protein requirement: \[27x + 16y \geq 80\]


• The budget constraint: \[3.20x + 1.75y \leq 10\]

To fully understand his situation, we also need to consider that Mark cannot buy negative amounts of protein water or bars, so we add:

• \[x \geq 0\]


• \[y \geq 0\]

When combined, these inequalities form a system that defines the range of possible purchases he can make.

graphing inequalities

Graphing inequalities is a visual way to represent the solutions to a system of inequalities. Each inequality corresponds to a region in the coordinate plane. For example:

• To graph \[27x + 16y \geq 80\], we first plot the line \[27x + 16y = 80\]. Then, we shade the region above this line since we need a value greater than or equal to 80.


• Next, we graph \[3.20x + 1.75y \leq 10\], by plotting the line \[3.20x + 1.75y = 10\] and shading the region below it to show the budget constraint.

We also include the non-negativity constraints. This confines the feasible region to the first quadrant (where both x and y are non-negative). The solution to the system is where these shaded regions overlap.

linear programming

Linear programming is an optimization technique for a system of linear inequalities with an objective function. In Mark's case, our goal isn't necessarily to optimize anything specific, but to find a feasible solution. The basic steps in linear programming involve:

• Formulating the problem: Define constraints and set up inequalities.


• Graphing the inequalities: Represent the constraints visually.


• Identifying the feasible region: The overlap of all shaded regions on the graph.
• Checking specific points: See if specific combinations satisfy all inequalities.

This method ensures whether a solution is possible given all constraints.

budget constraints

Budget constraints refer to the limitations on the amount of money that can be spent. Here’s how it applies to Mark:

• Each bottle of protein water costs \$3.20 and each protein bar costs \$1.75.


• Mark has \$10 to spend, which gives the inequality: \[3.20x + 1.75y \leq 10\]

When dealing with budget constraints in a system of inequalities, it’s crucial to ensure that the total expenditure does not exceed the available budget. Therefore, even if a combination of items meets the nutritional requirement, it must also respect the budget to be considered a feasible option.