Examkin

Question

Difficulty: MediumMatrices and Matrix Operations

A community theater group sells student tickets and adult tickets for their weekend plays. The number of tickets sold for Friday and Saturday night shows is represented by the matrix TT:

T=[80120100150]T = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix}

where the first and second rows represent Friday and Saturday, respectively, and the first and second columns represent student and adult tickets, respectively. The price of each ticket is represented by the matrix PP:

P=[610]P = \begin{bmatrix} 6 \\ 10 \end{bmatrix}

where the first row represents the price of a student ticket (6)andthesecondrowrepresentsthepriceofanadultticket(6) and the second row represents the price of an adult ticket ( 10). Which of the following matrices represents the total ticket sales revenue, in dollars, for Friday and Saturday, respectively?

  1. A
    [4801,2006001,500]\begin{bmatrix} 480 & 1,200 \\ 600 & 1,500 \end{bmatrix}
  2. B
    [2,1001,680]\begin{bmatrix} 2,100 \\ 1,680 \end{bmatrix}
  3. [1,6802,100]\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}Answer
  4. D
    [1,5802,100]\begin{bmatrix} 1,580 \\ 2,100 \end{bmatrix}
  5. E
    [1,5201,900]\begin{bmatrix} 1,520 \\ 1,900 \end{bmatrix}

Answer

The matrix with entries 1,680 and 2,100
The correct answer is the matrix with entries 1,680 and 2,100. This is found by multiplying the sales matrix by the price matrix. The revenue for Friday is calculated as 80×6+120×10=480+1,200=1,68080 \times 6 + 120 \times 10 = 480 + 1,200 = 1,680. The revenue for Saturday is 100×6+150×10=600+1,500=2,100100 \times 6 + 150 \times 10 = 600 + 1,500 = 2,100. Placing these in a column matrix matching the order of the days yields the correct matrix.

Step-by-Step Solution

1
Set up the matrix multiplication of the ticket sales matrix TT and the ticket price matrix PP.
TP=[80120100150][610]TP = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix} \begin{bmatrix} 6 \\ 10 \end{bmatrix}
To find the total revenue for Friday and Saturday, we must multiply the ticket sales by their respective prices.
2
Perform the multiplication by taking the dot product of each row of TT with the column of PP.
[80(6)+120(10)100(6)+150(10)]=[480+1200600+1500]\begin{bmatrix} 80(6) + 120(10) \\ 100(6) + 150(10) \end{bmatrix} = \begin{bmatrix} 480 + 1200 \\ 600 + 1500 \end{bmatrix}
The matrix product of a 2×22 \times 2 matrix and a 2×12 \times 1 matrix results in a 2×12 \times 1 matrix where the entries are the sums of the products of corresponding elements.
3
Simplify the sums in the resulting matrix.
[1,6802,100]\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}
Adding the products gives the final total revenues for Friday and Saturday, respectively.

Key Concept

Matrix Multiplication in Word Problems

Alternative Method

Instead of formal matrix multiplication, you can calculate the scalar totals for each day directly (Friday: 80×6+120×10=1,68080 \times 6 + 120 \times 10 = 1,680; Saturday: 100×6+150×10=2,100100 \times 6 + 150 \times 10 = 2,100) and match them with the corresponding rows of the resulting 2×12 \times 1 matrix.
Estimated Time:1m 30s
Rate this question

Topics

Matrices and Matrix Operations

More questions from this topic