Question

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Question 4 (Essay Worth 4 points)

The cost of attending an amusement park is $10 for children and $20 for adults. On a particular day, the attendance at the amusement park is 30,000 attendees, and the total money earned by the park is $500,000. Use the matrix equation to determine how many children attended the park that day. Use the given matrix equation to solve for the number of children’s tickets sold. Explain the steps that you took to solve this problem.

A matrix with 2 rows and 2 columns, where row 1 is 1 and 1 and row 2 is 10 and 20, is multiplied by matrix with 2 rows and 1 column, where row 1 is c and row 2 is a, equals a matrix with 2 rows and 1 column, where row 1 is 30,000 and row 2 is 500,000.

Solve the equation using matrices to determine the number of children's tickets sold. Show or explain all necessary steps.

Answer 1

The given matrix equation can be written as:

[1 1; 10 20] * [c; a] = [30,000; 500,000]

Multiplying the matrices on the left side of the equation gives us the system of equations:

c + a = 30,000 10c + 20a = 500,000

To solve for c and a using matrices, we can use the inverse matrix method. First, we need to find the inverse of the coefficient matrix [1 1; 10 20]. The inverse of a 2x2 matrix [a b; c d] can be calculated using the formula: (1/(ad-bc)) * [d -b; -c a].

Let’s apply this formula to our coefficient matrix:

The determinant of [1 1; 10 20] is (120) - (110) = 10. Since the determinant is not equal to zero, the inverse of the matrix exists and can be calculated as:

(1/10) * [20 -1; -10 1] = [2 -0.1; -1 0.1]

Now we can use this inverse matrix to solve for c and a. Multiplying both sides of our matrix equation by the inverse matrix gives us:

[2 -0.1; -1 0.1] * [c + a; 10c + 20a] = [2 -0.1; -1 0.1] * [30,000; 500,000]

Solving this equation gives us:

[c; a] = [25,000; 5,000]

So, on that particular day, there were 25,000 children’s tickets sold.