A banquet hall offers two types of tables for rent: 6...

$A$ banquet hall offers two types of tables for rent: 6 -person rectangular tables at a cost of $\$ 28$ each and 10-person round tables at a cost of \$52 each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The room can have a maximum of 35 tables and the hall only has 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

$A$ banquet hall offers two types of tables for rent: 6 -person rectangular tables at a cost of $\$ 28$ each and 10-person round tables at a cost of \$52 each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The room can have a maximum of
35 tables and the hall only has 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

Key Concepts

Linear Programming

A technique used to optimize an objective function, such as minimizing cost or maximizing profit, while satisfying a set of linear constraints. It involves decision variables, a linear objective, and a series of restrictions that the solution must adhere to.

Objective Function

The mathematical expression that represents the goal of the problem—in this case, minimizing cost. It is constructed as a linear combination of the decision variables, and its optimal value is determined subject to the constraints.

Constraints

These are the conditions or limitations imposed on the decision variables, typically represented as linear equations or inequalities. Constraints might include limitations on resources, capacity, or other requirements that must be satisfied for the solution to be feasible.

Feasible Region

The set of all points that satisfy the problem's constraints. It is the space within which the optimal solution must lie, and in linear programming, this region is typically a convex polygon where each vertex is a potential candidate for the optimal solution.

Corner Point Theorem

A principle stating that if an optimal solution exists for a linear programming problem, it will occur at a vertex (corner point) of the feasible region. This theorem is used to simplify the search for an optimal solution by only considering the vertices.

Transcript

00:01 All right, so we have a situation where we are planning a banquet and for a wedding and we need 250 people to be seated and we have two type tables we can choose from.

00:26 We have a rectangle table that seats six people and costs $28 and we have a round table that seats 10 people and costs $52 a piece.

00:37 The only other, we can't fit any more than 35 tables in the banquet hall and the only other constraint is that there are only 15 rectangular tables available so we can't go above 15 rectangular tables.

00:53 So we want to know how many rectangle tables and how many round tables minimize the cost and what that total minimum cost is.

01:02 So if we can, if we let x be the number of rectangular tables, then 6x is the number of people that we can seat at rectangular tables.

01:12 28x is the cost of the rectangular tables.

01:16 Then the round tables seat 10.

01:18 We're going to let y be the number of round tables that we get.

01:23 So 10 times y would be the people seated at the round tables and 52 times y would be the cost of the round tables.

01:33 So we've got 6x plus 10y representing the number of people and that has to seat, it has to be greater than or equal to 250.

01:44 And then the cost, this is our target function for our cost, so we're going to move that down here.

01:55 We know that the sum of the tables cannot exceed, has to be less than or equal to 35.

02:06 We also know that x has to be less than or equal to 15.

02:15 So those are our constraint functions.

02:18 So let's go ahead and put those in to our graphing tool.

02:22 6x plus 10y has to be greater than or equal to 250 because we have at least 250.

02:32 The number of tables total has to be less than or equal to 35 and we also know that x has to be less than 15.

02:44 So we're going to zoom out here and see where our overlapping shape is.

02:54 So we have three options it looks like.

03:01 There are three, four points we need to check.

03:04 We need to check, where is, x has to be less than 15.

03:10 So we don't need to check this point, x has to be less than 15.

03:14 So we can check 15 round and 16, 15 rectangular and 16 round.

03:22 We can check 15 rectangular and 20 round.

03:27 We can check 0 .25 and we can check 0 .35.

03:39 So those are our four options...

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