A catering service is organizing two events: a wedding and a corporate gala. The matrix P … — ACT

A catering service is organizing two events: a wedding and a corporate gala. The matrix $P$ represents the number of Deluxe packages ( $d$ ) and Premium packages ( $p$ ) ordered for each event:

P = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix}

where the rows represent the Wedding and the Gala respectively, and the columns represent Deluxe and Premium packages respectively.

The matrix $R$ represents the requirements (in units) of protein and vegetables for each package type:

R = \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix}

where the rows represent Deluxe and Premium packages respectively, and the columns represent protein and vegetables respectively.

The total units of ingredients required for each event are represented by the product matrix $T = P \times R$ .

If the total vegetables required for the wedding is 220 units, and the total protein required for the corporate gala is 560 units, what is the value of the sum $x + y$ ?

A
11
B
54.5
C
82.8
83Answer
E
145.5

Answer

The correct option is the value 83. Setting up the product matrix

T = P \times R

, the entry in row 1, column 2 (representing wedding vegetables) is

50(2) + 40y = 220

, which simplifies to

y = 3

. The entry in row 2, column 1 (representing gala protein) is

80(3) + 4x = 560

, which simplifies to

x = 80

. Adding these values together gives

x + y = 83

Step-by-Step Solution

Set up the matrix product expression

T = P \times R = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix} \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix} = \begin{bmatrix} 50(3) + 40(4) & 50(2) + 40(y) \\ 80(3) + 4x & 80(2) + xy \end{bmatrix}

We multiply row elements of the first matrix by column elements of the second matrix to obtain the entries of the product matrix.

Isolate the equation for vegetables required for the wedding

50(2) + 40y = 220 \implies 100 + 40y = 220 \implies 40y = 120 \implies y = 3

The wedding corresponds to the first row of matrix

P

, and vegetables correspond to the second column of matrix

R

. Their dot product equals the total vegetables for the wedding.

Isolate the equation for protein required for the corporate gala

80(3) + 4x = 560 \implies 240 + 4x = 560 \implies 4x = 320 \implies x = 80

The corporate gala corresponds to the second row of matrix

P

, and protein corresponds to the first column of matrix

R

. Their dot product equals the total protein for the gala.

Find the sum of the variables

x + y = 80 + 3 = 83

We add the solved values of

x

and

y

to obtain the requested sum.

Key Concept

Matrix Multiplication and Word Problem Translation

Estimated Time:3m 0s

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