Math Problem Statement
Set up a system of equations and then solve using inverse matrics.
A manufacturer of portable tools has three sets, Basic, Homeowner, and Pro, which must be painted, assembled, and packaged for shipping. The following table gives the number of hours required for each operation for each set.
Basic Homeowner Pro
Painting 1.1 1.8 2.4 Assembly 0.6 1.2 2 Packaging 1 1 1.4
If the manufacturer has 85.4 hours for painting per day, 58.8 hours for assembly per day, and 56.8 hours for packaging per day. How many sets of each type can be produced each day?
The manufacturer can produce Basic sets, Homeowner sets and
Pro sets per day.
Solution
We can solve this problem using a system of equations and the inverse matrix method.
Step 1: Define Variables
Let:
- = number of Basic sets produced per day
- = number of Homeowner sets produced per day
- = number of Pro sets produced per day
Step 2: Set Up the System of Equations
Using the given table, we set up equations for each type of operation:
Painting:
Assembly:
Packaging:
Thus, the system of equations is:
1.1x + 1.8y + 2.4z = 85.4 \\ 0.6x + 1.2y + 2z = 58.8 \\ 1x + 1y + 1.4z = 56.8 \end{cases}$$ ### **Step 3: Express in Matrix Form** This system can be written as: $$\begin{bmatrix} 1.1 & 1.8 & 2.4 \\ 0.6 & 1.2 & 2 \\ 1 & 1 & 1.4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 85.4 \\ 58.8 \\ 56.8 \end{bmatrix}$$ Let $$A = \begin{bmatrix} 1.1 & 1.8 & 2.4 \\ 0.6 & 1.2 & 2 \\ 1 & 1 & 1.4 \end{bmatrix} , \quad B = \begin{bmatrix} 85.4 \\ 58.8 \\ 56.8 \end{bmatrix}$$ To solve for $$X$$ (which contains $$x, y, z$$), we use the inverse matrix formula: $$X = A^{-1} B$$ ### **Step 4: Compute $$A^{-1}$$ and Solve** I will now compute the inverse of matrix $$A$$ and solve for $$X$$. numpy as np # Define matrix A (coefficients) and B (constants) A = np.array([[1.1, 1.8, 2.4], [0.6, 1.2, 2], [1, 1, 1.4]]) B = np.array([[85.4], [58.8], [56.8]]) # Compute the inverse of A A_inv = np.linalg.inv(A) # Solve for X (number of sets per day) X = np.dot(A_inv, B) X### **Final Answer** The manufacturer can produce per day: - **22 Basic** sets - **18 Homeowner** sets - **12 Pro** sets Would you like a step-by-step breakdown of how the inverse matrix was computed? ### **Related Questions** 1. How do you compute the inverse of a 3×3 matrix manually? 2. What are the conditions for a matrix to have an inverse? 3. How can matrix methods be applied to other manufacturing problems? 4. What are some real-world scenarios where solving systems using matrices is useful? 5. How does row reduction compare to using the inverse matrix method for solving systems? #### **Tip:** When solving systems of equations using matrices, always check if the determinant of the coefficient matrix is nonzero. If it is zero, the system may not have a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Linear Equations
Matrix Inversion
Algebra
Formulas
System of equations: Ax = B
Matrix Inversion: X = A^-1 * B
Theorems
Inverse Matrix Method
Cramer's Rule (optional alternative for solving systems)
Suitable Grade Level
Grades 11-12
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