Solve each problem. The Bread Box, a small neighborhood...

Solve each problem. The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix $A$. CAN'T COPY THE TABLE The cost (in cents) for each ingredient when purchased in either large lots or small lots is given in matrix $B$. CAN'T COPY THE TABLE (a) Use matrix multiplication to find a matrix giving the cost per item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Solve each problem.
The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix $A$.
CAN'T COPY THE TABLE
The cost (in cents) for each ingredient when purchased in either large lots or small lots is given in matrix $B$.
CAN'T COPY THE TABLE
(a) Use matrix multiplication to find a matrix giving the cost per item for the two purchase options.
(b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders.
(c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra where the rows of one matrix are combined with the columns of another to produce a new matrix. It is used to aggregate data from separate matrices, such as combining ingredients per item with cost data or orders with ingredient requirements. This process is instrumental in modeling and solving various real-world problems with multiple related variables.

Modeling Real-World Problems with Matrices

Using matrices to represent real-world scenarios involves organizing different data sets into structured arrays. In this context, matrices are used to represent ingredient requirements per bakery item, cost information for ingredients, and customer orders. This systematic representation allows for efficient computation and analysis, making it easier to derive meaningful insights through operations like matrix multiplication.

Linear Combinations

The concept of linear combinations entails forming new quantities by scaling and summing a set of variables. In the problem, each item’s total cost or ingredient usage is determined by multiplying corresponding entries in matrices and summing the results, effectively creating a linear combination that represents the aggregate cost or quantity.

Cost Analysis Through Matrices

Cost analysis in this setting involves constructing matrices that encapsulate cost per ingredient and ingredient usage per item. By applying matrix multiplication, one can analyze how different purchasing options impact overall costs. This approach demonstrates how algebraic techniques can be integrated into financial planning and operational optimization.

Transcript

00:01 All right, here we have question number 80.

00:04 It's a lot of simple math.

00:06 I have written out to make things a little bit smoother to go through.

00:10 But this question is, is they have a small neighborhood bakery that sells four items, sweet rolls, which are here just written as rolls, bread, cakes, and pies.

00:21 And so that is one giant matrix that has the amount of each ingredient, eggs, flour, sugar, and milk in those four, which makes up matrix a.

00:32 And then matrix b is the cost of making a large lot and a small lot for each ingredient.

00:40 So if you buy in bulk, nearly all the time, it's a little bit cheaper, but not all the time.

00:45 So i think all the time maybe.

00:48 Anyway, so starting with the first question, part a, it says to use matrix multiplication to find a matrix giving the cost per item of the two purchase orders.

00:57 So i just did the math here.

00:59 What you would do, say, for the first one, you have eggs and rules, and you have one egg per roll.

01:05 So for a small lot, it'll be $5, and for a large lot, it'll be $5, because there's no difference between those two.

01:13 Then if you move one over, for flour for rolls, there are four cups of flour per roll.

01:20 And the cost of a cup in a large lot is $8, so $4 times $8 is $32, but the cost of flour in a small lot is $10, so $4 times $10 .40.

01:32 So you have to go ahead and do all of that math.

01:34 Just easy simple math for all five ingredients across all of the different recipes, rolls, bread, cakes, and pies.

01:48 And so that's what you would do to make your two matrices.

01:51 So you could do a number of ways, but you could have one matrix here for the large cost, and then one matrix here for the small cost.

02:00 And you could put the individual ingredients into it, but really, really, that will be kind of a big matrix.

02:09 So an easier thing that i've done here is that i have just added up all of the costs of the different ingredients.

02:15 So i made the total large cost for rolls and the total small cost for rolls.

02:20 Same for all of these other recipes...

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