Math Problem Statement
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Problem 1:
It involves matrix multiplication to find total sales of refreshments and snacks over three months.
Matrix (sales quantities for each month) and matrix (unit prices for refreshments and snacks) are given as:
1500 & 2600 & 3650 \\ 750 & 800 & 900 \end{pmatrix} \quad B = \begin{pmatrix} 2.0 & 3.5 \\ 1.5 & 3.0 \\ 1.0 & 2.5 \end{pmatrix}$$ The questions are: - **(a)** Multiply the matrices to get total income from refreshments. Which element provides this information? - **(b)** Multiply to get total monthly income. In which month was the maximum income achieved? - **(c)** What was the total income over all three months? ### Steps: 1. Perform matrix multiplication $$C = A \times B$$. 2. Analyze the elements of the resulting matrix $$C$$. Let's solve it: $$C = \begin{pmatrix} 1500 & 2600 & 3650 \\ 750 & 800 & 900 \end{pmatrix} \cdot \begin{pmatrix} 2.0 & 3.5 \\ 1.5 & 3.0 \\ 1.0 & 2.5 \end{pmatrix}$$ I’ll compute this step by step.After performing the matrix multiplication, we get the following result: $$C = \begin{pmatrix} 10550 & 22175 \\ 3600 & 7275 \end{pmatrix}$$ Where: - The first row represents the income from **refreshments** for the months June, July, and August. - The second row represents the income from **snacks** for the same months. ### Now, let's address each part of the problem: #### (a) Multiply the matrices to obtain the income from selling refreshments in the three months. - The **first element** of the matrix $$C$$ (10550) corresponds to the income from **refreshments** in June. This information is given by the first element of the first row. - The income from **snacks** is represented by the second row of the matrix. #### (b) Which month had the maximum income? - Total income from all products for each month can be calculated by summing the corresponding elements from both rows of $$C$$: - June: $$10550 + 3600 = 14150$$ - July: $$22175 + 7275 = 29450$$ The month with the highest income is **July** with 29,450 euros. #### (c) What were the total sales across all three months? The total income from refreshments and snacks for the three months is the sum of all elements in matrix $$C$$: $$10550 + 22175 + 3600 + 7275 = 43500 \, \text{euros}.$$ --- Now, let's move on to **Problem 2**, which asks to find a function whose second derivative is $$-12x$$ and whose graph has a minimum at the point $$(-2,0)$$. Would you like to proceed with solving Problem 2, or do you have any questions about Problem 1? --- ### 5 related questions: 1. How do you interpret the elements of a matrix multiplication result in business contexts like this one? 2. How would the solution change if the prices increased by 10%? 3. Could you identify which matrix multiplication property applies here? 4. How can this type of calculation be automated for a large supermarket chain with hundreds of products? 5. How does matrix multiplication apply in fields like economics or engineering? ### Tip: In matrix multiplication, ensure that the dimensions of the matrices match properly, where the number of columns in the first matrix must equal the number of rows in the second matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Second Derivative
Probability
Formulas
Matrix Multiplication: A * B
Second Derivative: d^2y/dx^2 = -12x
Probability: P(A|B) = P(A and B)/P(B)
Theorems
Matrix multiplication rules
Differentiation and finding critical points
Bayes' Theorem
Suitable Grade Level
College or Advanced High School
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