Hi I am after some help starting this question.
A local hardware store sells three types of boxes; small, medium and large. The cost of the boxes are; small
$10, medium $20 and large $30. In the previous week a total of 1200 boxes were sold resulting in proceeds
of $20 000 for the week. The number of small boxes sold was equal to the sum of the medium and large
boxes sold.
Use matrix methods to determine how many of each type of box was sold.
To solve this problem using matrix methods, we can set up a system of equations.
Let's define the variables as follows:
- Let x represent the number of small boxes sold.
- Let y represent the number of medium boxes sold.
- Let z represent the number of large boxes sold.
Based on the given information, we can write the following equations:
Equation 1: The cost equation based on the total proceeds:
10x + 20y + 30z = 20000
Equation 2: The equation representing the total number of boxes sold:
x + y + z = 1200
Equation 3: The equation representing the relation between the number of small, medium, and large boxes sold:
x = y + z
We can rewrite Equation 3 as x - y - z = 0.
In matrix form, we have:
[10 20 30] [x] = [20000]
[ 1 1 1] [y] [1200]
[ 1 -1 -1] [z] [ 0]
To solve this system of equations using matrix methods, we can perform Gaussian elimination or use matrix inversion.
Let's use matrix inversion to solve the system of equations. We need to find the inverse of the coefficient matrix [A] and multiply it by the constant matrix [B]:
[A]⁻¹[B] = [x y z]
Let's set up the matrices:
[A] = [[10 20 30]
[ 1 1 1]
[ 1 -1 -1]]
[B] = [[20000]
[ 1200]
[ 0]]
To find the inverse of matrix [A], we can use any appropriate method such as row reduction or the adjugate method.
Once we have the inverse of matrix [A], we can multiply it by matrix [B] to find the values of x, y, and z, representing the number of each type of box sold.
To solve this question using matrix methods, we can define a matrix equation.
Let's represent the number of small, medium, and large boxes sold as variables: s, m, and l, respectively.
Based on the given information, we can set up the following equations:
1) s + m + l = 1200 (total number of boxes sold)
2) 10s + 20m + 30l = 20000 (total proceeds from box sales)
Now we can convert these equations into a matrix equation.
Let's define a coefficient matrix, A, and a variable matrix (or vector), X, as follows:
A = [1 1 1]
X = [s]
[m]
[l]
The matrix equation can now be written as:
AX = B,
where B is the column matrix containing the right-hand sides of the equations:
B = [1200]
[20000]
To solve for X, we need to find the inverse of matrix A:
A^-1 * AX = A^-1 * B
Simplifying, we find:
X = A^-1 * B
By calculating the inverse of A and multiplying it by B, we can determine the values of s, m, and l, which represent the number of small, medium, and large boxes sold, respectively.