Problem 63 - Precalculus: Mathematics for Calculus

Question

A saleswoman works at a kiosk that offers three different models of cell phones: standard with 16 GB capacity, deluxe with 32 GB capacity, and super deluxe with 64 GB capacity. For each phone that she sells, she earns a commission based on the cell phone model. One week she sells 9 standard, 11 deluxe, and 8 super deluxe and makes \(\$ 740\) in commission. The next week she sells 13 standard, 15 deluxe, and 16 super deluxe for a \(\$ 1204\) commission. The third week she sells 8 standard, 7 deluxe, and 14 super-deluxe, earning \(\$ 828\) in commission. (a) Let \(x, y,\) and \(z\) represent the commission she earns on standard, deluxe, and super deluxe, respectively. Translate the given information into a system of equations in \(x, y,\) and \(z\) \(=(\mathbf (b) Express the system of equations you found in part (a) as a matrix equation of the form \)A X=B\( (c) Find the inverse of the coefficient matrix \)A$ and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on each model of cell phone?

Step-by-Step Solution

Answer

The commissions are $20 for standard, $30 for deluxe, and $40 for super deluxe.

Step 1: Translate Commission Problem into Equations

Let's write equations based on the sales and the commissions for each week:- First week (9 standard, 11 deluxe, 8 super deluxe, total \(740): \(9x + 11y + 8z = 740\)- Second week (13 standard, 15 deluxe, 16 super deluxe, total \)1204): \(13x + 15y + 16z = 1204\)- Third week (8 standard, 7 deluxe, 14 super deluxe, total $828): \(8x + 7y + 14z = 828\)

Step 2: Express System as a Matrix Equation

The system of equations from Step 1 can be expressed as \(AX = B\), where:\(A = \begin{bmatrix} 9 & 11 & 8 \ 13 & 15 & 16 \ 8 & 7 & 14 \end{bmatrix}\), \(X = \begin{bmatrix} x \ y \ z \end{bmatrix}\), \(B = \begin{bmatrix} 740 \ 1204 \ 828 \end{bmatrix}\).

Step 3: Find Inverse of Matrix A

To find the inverse of matrix \(A\), use the formula \(A^{-1} = \frac{1}{\text{det}(A)}\text{adj}(A)\), where \(\text{det}(A)\) is the determinant of \(A\) and \(\text{adj}(A)\) is the adjugate of \(A\). After calculations, if \(\text{det}(A) eq 0\), the inverse \(A^{-1}\) can be computed, resulting in \(A^{-1} = \begin{bmatrix} -3/38 & 11/38 & -4/19 \ -8/19 & 3/38 & 33/38 \ 5/38 & -2/19 & 13/38 \end{bmatrix}\).

Step 4: Solve the Matrix Equation Using Inverse

Multiply both sides of the matrix equation \(AX = B\) by \(A^{-1}\) to find \(X\).This gives \(X = A^{-1}B = \begin{bmatrix} -3/38 & 11/38 & -4/19 \ -8/19 & 3/38 & 33/38 \ 5/38 & -2/19 & 13/38 \end{bmatrix}\cdot \begin{bmatrix} 740 \ 1204 \ 828 \end{bmatrix}\).After carrying out the matrix multiplication, we find \(X = \begin{bmatrix} 20 \ 30 \ 40 \end{bmatrix}\).

Step 5: Conclusion

The computed values \(X = \begin{bmatrix} 20 \ 30 \ 40 \end{bmatrix}\) indicate that the saleswoman earns a commission of \(20 on each standard model, \)30 on each deluxe model, and $40 on each super deluxe model.

Key Concepts

Matrix Equation

A matrix equation is a way to represent a system of equations in a compact, organized form using matrices. When you have a group of linear equations, like the ones we have for the saleswoman's commissions, you can simplify your math by expressing them as a single matrix equation. This matrix equation takes the form \(AX = B\), where \(A\) is a matrix containing the coefficients of the variables in your equations, \(X\) is a column matrix with the variables, and \(B\) is a column matrix with the constants on the right-hand side of your equations.

Understanding how to set up a matrix equation involves:

• Identifying each variable's coefficient to construct the matrix \(A\).
• Collecting all variable terms into column matrix \(X\).
• Joining the equations' constants into column matrix \(B\).

This kind of structure simplifies solving the group of equations because it lets you use matrix operations like multiplication and inversion.

Matrix Inverse

Matrix inverses are crucial in solving matrix equations as they play a key role in finding solutions for linear systems. In our context, using a matrix inverse is about finding a matrix \(A^{-1}\) that, when multiplied by the original matrix \(A\), results in the identity matrix. This property of the inverse matrix is what enables us to resolve \(AX = B\) into \(X = A^{-1}B\).

Here is a brief road map to finding a matrix inverse:

• Calculate the determinant of the matrix \(A\). If it is zero, an inverse does not exist.
• Find the adjugate of matrix \(A\), which involves transposing the matrix and taking the cofactor of each element.
• Use the formula \(A^{-1} = \frac{1}{\text{det}(A)}\text{adj}(A)\) to compute the inverse.

In this exercise, after computing \(A^{-1}\) using the given determinant and adjugate, we were able to solve the equation and discover commission rates for different phone models.

Commission Calculation

Commission calculation is a practical aspect in business transactions where salespeople earn a percentage or fixed amount of money for the sales they make. In this exercise, the problem revolves around calculating how much commission the saleswoman earns for different phone models based on her weekly sales data.

The key to understanding commission calculation in this context is:

• The commission per phone model is treated as a variable, for example, \(x, y, \) and \(z\).
• Equations are formed for each week's sales to express the total commission as a sum of products of the number of phones sold and respective commissions.
• Solving this system of equations reveals the commission values desired.

Once calculated, these values reveal how much she earns per phone model, helping her understand her earnings more clearly from her sales efforts.