Problem 557 - Advanced Calculus

Question

Suppose a student has \(\$ 90\) with which to buy lecture notes at \(\$ 3\) each and packs of beer, at \(\$ 5\) each. Let the function $$ f(x, y)=x y $$ (1) express how much satisfaction is derived from buying \(\mathrm{x}\) lecture notes and \(\mathrm{y}\) packs of beer. What \(\mathrm{x}\) and \(\mathrm{y}\) will maximize the student's pleasure?

Step-by-Step Solution

Answer

The student should buy 5 lecture notes and 15 packs of beer to maximize their satisfaction, resulting in a satisfaction value of 75.

Step 1: Identify the constraints

As the student has $90 to spend on lecture notes and packs of beer, we can write the constraint as: \(3x + 5y \leq 90\)

Step 2: Find the critical points

To find the critical points, we can graph the constraint to see the feasible region. Then, we will find the points where the constraint function is equal to the maximum value within the feasible region: \(3x + 5y = 90\)

Step 3: Check for integer solutions

As the student can only buy whole quantities of lecture notes and packs of beer, we will look for integer solutions in the feasible region. This implies x and y will be non-negative integers. We can write the constraint equation as: \(y = \frac{90 - 3x}{5}\) Now, we would find the values of x and y in the form of non-negative integers that satisfy the constraint.

Step 4: Analyze the possible solutions

We will now look at different integer values of x to find plausible integer values for y. For x=0: \(y = \frac{90 - 3(0)}{5} = 18\) (integer value) For x=1: \(y = \frac{90 - 3(1)}{5} = 17.4\) (non-integer value) For x=2: \(y = \frac{90 - 3(2)}{5} = 16.8\) (non-integer value) For x=3: \(y = \frac{90 - 3(3)}{5} = 16.2\) (non-integer value) For x=4: \(y = \frac{90 - 3(4)}{5} = 15.6\) (non-integer value) For x=5: \(y = \frac{90 - 3(5)}{5} = 15\) (integer value) For x=6: \(y = \frac{90 - 3(6)}{5} = 14.4\) (non-integer value) We find that the solutions (x, y) are (0, 18) and (5, 15).

Step 5: Test the solutions and find the maximum satisfaction

We will plug in the feasible integer solutions into the satisfaction function \(f(x, y) = x y\) and compare the results: For (x, y) = (0, 18): \(f(0, 18) = 0\) For (x, y) = (5, 15): \(f(5, 15) = 5 * 15 = 75\) Comparing these values, we can conclude that buying 5 lecture notes and 15 packs of beer will maximize the student's pleasure with a satisfaction of 75. This means that the student should choose to buy(x, y) = (5, 15) to maximize their satisfaction.

Key Concepts

Constraint Satisfaction

In optimization problems, satisfaction constraints are the conditions that must be met for a solution to be considered valid. In this exercise, the constraint is the student's budget for buying lecture notes and packs of beer. The student's total spending must not exceed \(90.

This is expressed by the inequality:

• \(3x + 5y \leq 90\)

Here, \(x\) represents the number of lecture notes, and \(y\) represents the packs of beer. Each lecture note costs \\)3, and each pack of beer costs \$5. The constraint reflects the requirement to stay within budget, ensuring the proper application of limited resources. Ensuring we meet this condition is essential.
Every potential solution—combination of \(x\) and \(y\)—that satisfies this budget constraint is considered feasible.

Integer Solutions

In many real-world scenarios, especially in problems such as purchasing goods, only whole numbers make sense. You can't buy a fraction of a lecture note or half a pack of beer. Hence, we're interested in integer solutions for \(x\) and \(y\).

In our context:

• Both \(x\) and \(y\) must be non-negative integers.

After setting up our constraint equation, \(y = \frac{90 - 3x}{5}\), we must choose integer values for \(x\) such that \(y\) also becomes an integer.
Not all values of \(x\) will yield an integer \(y\). For example, when \(x = 1\), \(y = 17.4\), which is not feasible in this problem. Thus, only the combinations that produce integer results are further considered in the search for optimal solutions.

Maximum Satisfaction

Once we have the feasible integer solutions, the next step is to determine which combination of lecture notes and beer pack purchases maximizes satisfaction. The given satisfaction function is:

• \(f(x, y) = xy\)

This function reflects pleasure from acquiring the combination of \(x\) lecture notes and \(y\) beer packs.

Using the feasible integer solutions from the previous section, we test each by plugging them into this function to see which yields the highest value.
For instance, the solution \((5, 15)\) gives:

• \(f(5, 15) = 75\)

Whereas, the solution \((0, 18)\) results in \(f(0, 18) = 0\). Clearly, \(75\) is greater than \(0\), indicating that purchasing 5 lecture notes and 15 packs of beer provides maximum satisfaction under the given constraints.

Feasible Region

Visualizing solutions is often made easier by considering the concept of a feasible region, which is the set of all possible points (combinations of \(x\) and \(y\)) that satisfy the constraints. For the budget constraint \(3x + 5y \leq 90\), the feasible region can be depicted as an area on a graph where the inequality holds true. This area is bounded by the budget constraint line and the axes of the graph, since both \(x\) and \(y\) must be non-negative.

Within this region, we're looking specifically for integer points because, as we discussed, only whole numbers of goods can be purchased.

• Examples of feasible points include \((0, 18)\) and \((5, 15)\).

The feasible region simplifies the process of checking possible points and helps visualize where optimal solutions may lie. The task is, then, to scour these integer points within the region to find the one that provides the maximum satisfaction.
This method significantly decreases the potential combinations we need to evaluate, focusing only on viable options.