Computing the cost of Production The Acme Steel Company is a...

Computing the cost of Production The Acme Steel Company is a producer of stainless steel and aluminum containers. On a certain day, the following stainless steel containers were manufactured: 500 with 10 -gallon (gal) capacity, 350 with 5-gal capacity, and 400 with 1-gal capacity. On the same day, the following aluminum containers were manufactured: 700 with 10-gal capacity, 500 with 5-gal capacity, and 850 with 1-gal capacity. (a) Find a 2 by 3 matrix representing these data. Find a 3 by 2 matrix to represent the same data. (b) If the amount of material used in the 10 -gal containers is 15 pounds (lb), the amount used in the 5-gal containers is 8 lb, and the amount used in the 1-gal containers is 3 lb, find a 3 by 1 matrix representing the amount of material used. (c) Multiply the 2 by 3 matrix found in part (a) and the 3 by 1 matrix found in part (b) to get a 2 by 1 matrix showing the day's usage of material. (d) If stainless steel costs Acme $\$ 0.10$ per pound and aluminum costs $\$ 0.05$ per pound, find a 1 by 2 matrix representing cost. (e) Multiply the matrices found in parts (c) and (d) to find the total cost of the day's production.

Computing the cost of Production The Acme Steel Company is a producer of stainless steel and aluminum containers. On a certain day, the following stainless steel containers were manufactured: 500 with 10 -gallon (gal) capacity, 350 with 5-gal capacity, and 400 with 1-gal capacity. On the same day, the following aluminum containers were manufactured:
700 with 10-gal capacity, 500 with 5-gal capacity, and 850 with 1-gal capacity.
(a) Find a 2 by 3 matrix representing these data. Find a 3 by 2 matrix to represent the same data.
(b) If the amount of material used in the 10 -gal containers is 15 pounds (lb), the amount used in the 5-gal containers is 8 lb, and the amount used in the 1-gal containers is 3 lb, find a 3 by 1 matrix representing the amount of material used.
(c) Multiply the 2 by 3 matrix found in part (a) and the 3 by 1 matrix found in part (b) to get a 2 by 1 matrix showing the day's usage of material.
(d) If stainless steel costs Acme $\$ 0.10$ per pound and aluminum costs $\$ 0.05$ per pound, find a 1 by 2 matrix representing cost.
(e) Multiply the matrices found in parts (c) and (d) to find the total cost of the day's production.

Key Concepts

Matrix Representation

This concept involves organizing and structuring data into rows and columns to form a matrix. In many real-world applications, data is more manageable and interpretable when placed into a matrix, allowing for clear visualization of relationships between different variables, such as quantities produced, material usage, or other relevant attributes.

Matrix Multiplication

Matrix multiplication is a key operation where two matrices are combined to form a new matrix based on the sum of products of corresponding elements. This operation requires that the number of columns in the first matrix equals the number of rows in the second matrix, making it essential for calculating aggregated values and transforming data from one representation to another in applied problems.

Dimensional Compatibility

This concept refers to the requirement that for matrix multiplication to be valid, the dimensions of the matrices must align appropriately. Specifically, the number of columns in the first matrix must equal the number of rows in the second matrix. This principle is critical to ensuring that operations such as multiplying production matrices by material usage or cost matrices yield meaningful, correctly dimensioned results.

Matrix Modeling in Cost Computation

Matrix modeling in cost computation involves representing production, consumption, and cost data in matrices. By doing so, various elements of a production process—like quantities produced, resource requirements, and unit costs—can be combined using matrix operations to efficiently calculate overall costs. This approach simplifies the process of aggregating multiple factors into a single total cost figure.

Transcript

00:01 So in this problem, we're the acme steel company.

00:10 And what do we see? well, we see that on one day, if i were to make a list here of, say, the capacity of containers that are made, and the count or number of them, well, we made 500 of 10 gallon.

00:44 They're all in gallons so i can just do it this way.

00:48 10 gallons.

00:50 Capacity here is in gallons.

00:53 That's kind of ugly.

00:55 Let me fix that up.

00:59 Galens, okay.

01:01 We made 350 -five -gallon ones and 4001 gallon capacities.

01:13 Okay.

01:14 And then on another day, this is day one, day two count.

01:31 We made 700 of the 10 gallon ones, 500 of the 5 gallon ones, and 850 of the 1 gallon ones.

01:44 All right.

01:45 And so the first thing we're asked to do is find a 2x3 matrix representing these data.

01:51 And then we're asked to find a 3 by 2 matrix to represent the same data.

01:55 All right, so i have the data here.

01:59 So the first one, the 2 by 3 could look like this, couldn't it? two columns and three rows.

02:15 And then the other one, well, we would just switch this around, wouldn't we? we would go 500, 350, 400, 700, 500, 500, 500, 850.

02:35 Do them like that.

02:37 Okay.

02:40 The next question says that the amount of material used in 10 gallon containers is 15 pounds.

02:45 So we'll put the pounds here.

02:49 So 10 gallon containers is 15 pounds.

02:53 And the amount used in the five gallon containers is eight pounds the amount used in the one gallon containers is three pounds find a three by one matrix representing the amount of material used well, we just write those across in a row don't we 15 eight and three like so alright, so now we want to multiply the two by three i don't know hang on i got the wrong one two by three and the 3 by 1.

03:42 Okay.

03:43 So we need to go back here and fix this for a minute.

03:47 So i did this wrong, right? i need to make a couple of notes here...

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