T-shirt profits A clothing company makes a profit of 10 on...

T-shirt profits A clothing company makes a profit of $\$ 10$ on its long-sleeved T-shirts and $\$ 5$ on its short-sleeved T-shirts. Assuming there is a $\$ 200$ setup cost, the profit on T-shirt sales is $z=10 x+5 y-200,$ where $x$ is the number of long-sleeved T-shirts sold and $y$ is the number of short-sleeved T-shirts sold. Assume $x$ and $y$ are nonnegative. a. Graph the plane that gives the profit using the window $$[0,40] \times[0,40] \times[-400,400]$$ b. If $x=20$ and $y=10,$ is the profit positive or negative? c. Describe the values of $x$ and $y$ for which the company breaks even (for which the profit is zero). Mark this set on your graph.

T-shirt profits A clothing company makes a profit of $\$ 10$ on its long-sleeved T-shirts and $\$ 5$ on its short-sleeved T-shirts. Assuming there is a $\$ 200$ setup cost, the profit on T-shirt sales is $z=10 x+5 y-200,$ where $x$ is the number of long-sleeved T-shirts sold and $y$ is the number of short-sleeved T-shirts sold. Assume $x$ and $y$ are nonnegative.
a. Graph the plane that gives the profit using the window $$[0,40] \times[0,40] \times[-400,400]$$
b. If $x=20$ and $y=10,$ is the profit positive or negative?
c. Describe the values of $x$ and $y$ for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Key Concepts

Constrained Domains in Optimization

In many real-world problems, such as production scenarios, the variables are subject to restrictions like non-negativity. Recognizing and applying these constraints ensures that the mathematical solution is applicable and meaningful in a practical context, avoiding unrealistic values like negative quantities.

Three-Dimensional Graphing of Linear Equations

This involves visualizing and interpreting linear equations as planes in a three-dimensional space. Here, each axis represents a variable, and the plane shows all solutions that satisfy the equation. It helps in understanding how changes in one variable affect the outcome in multi-variable systems.

Linear Profit Functions

This concept refers to expressing profit as a linear combination of variables, which typically includes contributions from different products or activities and a fixed cost. It involves understanding how individual components (like revenue per item) combine with fixed expenses to yield a net profit, which is a fundamental notion in algebra and business mathematics.

Break-even Analysis

Break-even analysis is the process of determining the point where total revenues equal total costs, resulting in zero profit. This analysis is critical in business for determining the minimum performance needed to avoid losses, and it is achieved by setting the profit function equal to zero and solving for the variable conditions.

Transcript

Please give Ace some feedback