Solve each exercise by using the inverse of the coeffcient matrix 

to solve a system of equations.

63. Analysis of Orders  The Bread Box Bakery sells three types 

of cakes, each requiring the amounts of the basic ingredients 

shown in the following matrix.

Type of Cake

I  II  III

Ingredient

Flour 1in cups2

Sugar 1in cups2

Eggs

£

2

2

2

4

1

1

2

2

3

§

To fll its daily orders for these three kinds of cake, the bakery 

uses 72 cups of four, 48 cups of sugar, and 60 eggs.

(a)  Write a 3*1 matrix for the amounts used daily.

(b)  Let the number of daily orders for cakes be a 3*1 matrix 

X with entries x1

, x2

, and x3

. Write a matrix equation that 

can be solved for X, using the given matrix and the matrix 

from part (a).

(c)  Solve the equation from part (b) to fnd the number of 

daily orders for each type of cake.

We are given a system of equations related to the bakery's daily orders for three types of cakes. The bakery uses specific amounts of flour, sugar, and eggs for each type of cake, and we need to determine the number of daily orders for each cake using matrix methods.

### Part (a): Write a \( 3 \times 1 \) Matrix for the Amounts Used Daily

The bakery uses:
- 72 cups of flour,
- 48 cups of sugar,
- 60 eggs.

We can represent this as a \( 3 \times 1 \) matrix:

\[
\mathbf{B} = \begin{pmatrix} 72 \\ 48 \\ 60 \end{pmatrix}
\]

### Part (b): Write a Matrix Equation

Let the number of daily orders for the three types of cakes (Cake I, Cake II, and Cake III) be represented by the matrix:

\[
\mathbf{X} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}
\]

Where:
- \( x_1 \) is the number of orders for Cake I,
- \( x_2 \) is the number of orders for Cake II,
- \( x_3 \) is the number of orders for Cake III.

The given matrix of ingredients required for each cake is:

\[
A = \begin{pmatrix} 2 & 2 & 2 \\ 4 & 1 & 1 \\ 2 & 2 & 3 \end{pmatrix}
\]

Where each entry corresponds to the amount of each ingredient (flour, sugar, eggs) required for each type of cake.

The matrix equation that can be solved for \( \mathbf{X} \) is:

\[
A \mathbf{X} = \mathbf{B}
\]

Substituting the known values:

\[
\begin{pmatrix} 2 & 2 & 2 \\ 4 & 1 & 1 \\ 2 & 2 & 3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 72 \\ 48 \\ 60 \end{pmatrix}
\]

### Part (c): Solve the Matrix Equation

To solve for \( \mathbf{X} \), we first need to check if the matrix \( A \) is invertible. If it is, we can solve for \( \mathbf{X} \) using the formula:

\[
\mathbf{X} = A^{-1} \mathbf{B}
\]

Let’s calculate the inverse of matrix \( A \) and solve for \( \mathbf{X} \).The solution for the number of daily orders for each type of cake is:

\[
\mathbf{X} = \begin{pmatrix} 4 \\ 44 \\ -12 \end{pmatrix}
\]

This means:
- \( x_1 = 4 \) orders for Cake I,
- \( x_2 = 44 \) orders for Cake II,
- \( x_3 = -12 \), which doesn't make sense physically, as negative orders are not possible.

This negative value suggests that either there is an error in the data provided, or the bakery's supply constraints cannot be met with this combination of cakes. A revision of the problem might be necessary.

Let me know if you'd like to explore this further or make adjustments to the problem.

Here are 5 related questions to consider:
1. What does it mean if a system of equations yields a negative solution?
2. How do we interpret cases where the system has no realistic solutions?
3. Can we modify the constraints to achieve non-negative orders for each cake type?
4. How do we handle systems with more constraints than variables?
5. What real-world situations might lead to similar infeasibility in optimization problems?

**Tip:** When solving real-world problems with systems of equations

Solve each exercise by using the inverse of the coefficient matrix to solve a system of equations.

63. Analysis of Orders The Bread Box Bakery sells three types of cakes, each requiring the amounts of the basic ingredients shown in the following matrix.

Type of Cake
I  II  III

Ingredient
Flour (in cups)
Sugar (in cups)
Eggs

To fill its daily orders for these three kinds of cake, the bakery uses 72 cups of flour, 48 cups of sugar, and 60 eggs.

(a) Write a 3*1 matrix for the amounts used daily.
(b) Let the number of daily orders for cakes be a 3*1 matrix X with entries x1, x2, and x3. Write a matrix equation that can be solved for X, using the given matrix and the matrix from part (a).
(c) Solve the equation from part (b) to find the number of daily orders for each type of cake.

Learn how to solve a system of equations using matrix inversion for a bakery's daily orders of three types of cakes. This example involves applying linear algebra concepts to compute daily orders based on the bakery's ingredient usage. Suitable for Grades 11-12, this lesson covers systems of equations and matrix inversions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation