Refer to Problem 54 . The company makes a profit of 50...

Refer to Problem $54 .$ The company makes a profit of 50 dollars on each table and a profit of 15 dollars on each chair. (A) If the company makes 20 tables and 20 chairs per day, the daily profit will be 1,300 dollars. Are there other feasible production schedules that will result in a daily profit of 1,300 dollars? How are these schedules related to the graph of the line $50 x+15 y=1,300 ?$ (B) Find a feasible production schedule that will produce a daily profit greater than 1,300 dollars and repeat part A for this schedule. (C) Discuss methods for using lines like those in parts $\mathrm{A}$ and $\mathrm{B}$ to find the largest possible daily profit.

Refer to Problem $54 .$ The company makes a profit of 50 dollars on each table and a profit of 15 dollars on each chair.
(A) If the company makes 20 tables and 20 chairs per day, the daily profit will be 1,300 dollars. Are there other feasible production schedules that will result in a daily profit of 1,300 dollars? How are these schedules related to the graph of the line $50 x+15 y=1,300 ?$
(B) Find a feasible production schedule that will produce a daily profit greater than 1,300 dollars and repeat part A for this schedule.
(C) Discuss methods for using lines like those in parts $\mathrm{A}$ and $\mathrm{B}$ to find the largest possible daily profit.

Key Concepts

Linear Equation Modeling

This concept involves representing relationships between quantities—such as production counts and profit—using linear equations. In production scenarios, the profit function is formulated as a linear equation where each term represents the contribution of a product to the total profit, allowing one to analyze how different combinations of production levels affect the overall outcome.

Graphical Analysis of Linear Equations

Graphing a linear equation in two variables provides a visual interpretation of all the combinations that satisfy the equation. The line represents all possible production schedules that yield a fixed profit, with each point corresponding to a specific combination of products produced, which helps in understanding the interplay between different variables in the equation.

Feasible Production Schedules and Constraints

In the context of production planning, feasible production schedules refer to combinations of product outputs that meet the required constraints, both in terms of resources and targets such as profit. Identifying these schedules involves evaluating the linear relationship and ensuring that the solutions are practical, often considering restrictions like integer values for discrete units of production.

Optimization and Linear Programming Techniques

This concept focuses on methods to determine the best possible outcome—such as the maximum profit—under given constraints. Linear programming uses a combination of algebraic and graphical methods to optimize an objective function (in this case, profit), navigating through the feasible region defined by the linear equations or inequalities to identify the optimal production schedule.

Transcript

00:01 Okay, so this question asks us to go back to a previous question, and problem 54, and so these are the inequalities from that problem.

00:09 And we're dealing with tables and chairs, and the tables are going to give us a, we're going to do with profits.

00:14 Tables are going to give us a profit at $50, and chairs $15.

00:18 So if i set up a profit equation, it will be 50 times x plus 15 times y gives me the profit.

00:26 And then the problem tells us that if i do 20 and 20, 20 tables and 20 chairs, i'll get a profit of, $1 ,300.

00:32 So 20 tables and 20 chairs gives me $1 ,300.

00:39 So then i'm going to set up a line that tells me how i can make $1 ,300.

00:46 We know that 20 and 20 will work because they've given us that.

00:49 But this line is going to represent all possible solutions to reach $1 ,300.

00:54 Okay.

00:55 And so if i go to my graph here, here's the blue line that's going to give me everything to $1 ,300.

01:04 And so any, i have to be underneath the purple and the red lines.

01:08 And so anything from here to here is going to give me, on that line is going to give me a profit of $1 ,300.

01:13 And so i'm going to go ahead and just use this intercept, $2 ,000.

01:16 So if i do 26 tables and zero chairs, i will get a profit of $1 ,300.

01:24 Because 26 times the 50 is going to give me the $1 ,300.

01:29 Okay, so that's one possible, for part a here, that's one possible production schedule that will give me a profit of $1 ,300.

01:38 Now it asks us to find a production schedule that will produce profit greater.

01:41 So i'm just going to go ahead and find a spot above the blue line.

01:49 And so if i want to find a spot that's above the blue line that still works, i can go ahead with.

01:59 So let's go ahead and choose a lot of...

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