Problem 61 - College Algebra with Modeling and Visualization

Question

Raising Animals \(\mathbf{A}\) breeder can raise no more than 50 hamsters and mice and no more than 20 hamsters. If she sells the hamsters for \(\$ 15\) each and the mice for \(\$ 10\) each, find the maximum revenue produced.

Step-by-Step Solution

Answer

The maximum revenue is \(\$600\), achieved by selling 20 hamsters and 30 mice.

Step 1: Define Variables

Let \( h \) represent the number of hamsters and \( m \) represent the number of mice. The total number of hamsters and mice cannot exceed 50, giving us the constraint: \( h + m \leq 50 \). Likewise, the maximum number of hamsters is 20, so: \( h \leq 20 \).

Step 2: Objective Function

The revenue \( R \), which we want to maximize, is given by the earnings from selling hamsters and mice. Thus, the objective function is: \( R = 15h + 10m \).

Step 3: Identify Constraints

We have the following constraints:1. \( h + m \leq 50 \) (Total animals constraint)2. \( h \leq 20 \) (Hamster constraint)3. \( h, m \geq 0 \) (Non-negativity constraint)

Step 4: Solve the System of Inequalities

Graph the inequalities on a coordinate plane. The feasible region is the area that satisfies all constraints. We need to evaluate the objective function at each vertex of this region.

Step 5: Calculate Revenue at Vertices

The vertices of the feasible region are:- \((0,0)\) with \( R = 15(0) + 10(0) = 0\)- \((0,50)\) with \( R = 15(0) + 10(50) = 500\)- \((20,30)\) with \( R = 15(20) + 10(30) = 600\)- \((20,0)\) with \( R = 15(20) + 10(0) = 300\)

Step 6: Determine Maximum Revenue

By comparing the revenues calculated at each vertex, the maximum revenue is \( R = 600 \) at the point \((20,30)\), meaning the breeder should raise 20 hamsters and 30 mice.

Key Concepts

Objective Function

In the context of linear programming, the objective function is integral because it is what we aim to optimize—either by maximizing or minimizing its value. For the problem of raising animals, our goal is to maximize revenue, which is the financial return. The revenue is computed based on the number of items sold, at specific prices.
Using the given prices, we define the objective function as:

• For each hamster sold, the breeder earns \\(15. Therefore, if the breeder sells \( h \) hamsters, the contribution to the revenue from hamsters is \( 15h \).
• For each mouse sold, the breeder earns \\)10. Thus, the revenue from selling \( m \) mice is \( 10m \).

Putting it together, the total revenue function, which is our objective function, can be expressed as:\[R = 15h + 10m\]
This function provides a clear relationship between the number of animals sold and the total revenue, guiding us in choosing the optimal number of each animal to maximize earnings.

System of Inequalities

A system of inequalities helps define the feasible solutions to a linear programming problem. Each inequality sets limits or conditions the variables must satisfy. These inequalities carve out a region on a graph known as the feasible region, where the solution can be found.
For the breeder:

• The constraint \( h + m \leq 50 \) ensures that the total number of animals does not exceed 50.
• The constraint \( h \leq 20 \) signifies that no more than 20 hamsters can be raised.
• Non-negativity constraints \( h, m \geq 0 \) ensure that a negative number of animals is not possible.

These constraints are represented by lines on a graph. The area where all these constraints overlap is the feasible region. Any solution within this region satisfies all given conditions and is a potential candidate for optimizing the objective function.

Constraints

Constraints are the limits within which a problem must be solved. They are essential in linear programming as they define the boundaries for the optimal solution.
In this exercise:

• The **total animals constraint** \( h + m \leq 50 \) ensures the breeder doesn't exceed the pen's capacity.
• The **hamster constraint** \( h \leq 20 \) restricts the number of hamsters, reflecting any additional limiting factors like space or resources.
• The **non-negativity constraint** ensures we can't have negative animals, so \( h, m \geq 0 \).

Constraints ensure that the solutions proposed are realistic and practical. They are critical in finding a feasible region from which the objective function is then optimized. Each constraint plays a role in shaping the feasible region, impacting where the maximum (or minimum) of the objective function can be found.