Question

The Collin Canoe Company makes three types of inflatable canoes

answer all parts

Answer 1

a.

x=20

y=220

z=100

b.

0.5 1 1.5 380

0.6 0.9 1.2 330

0.2 0.3 0.5 120

c.

5 10 15 / 3800

6 9 12 / 3300

2 3 5 / 1200

Answer 2

a. Variables:
`\( x \)` is one-person canoes,
`\( y \)` is two-person canoes,
`\( z \)` is four-person canoes.

b. Filled Table:

( Given Below)

c. Augmented Matrix:

`\[\begin{bmatrix}0.5 & 1 & 1.5 & 380 \\0.6 & 0.9 & 1.2 & 330 \\0.2 & 0.3 & 0.5 & 120 \\\end{bmatrix}\]`

d. RREF of the matrix:

`\[\begin{bmatrix}1 & 0 & 0 & 20 \\0 & 1 & 0 & 220 \\0 & 0 & 1 & 100 \\\end{bmatrix}\]`

e. Final answer: To be at full capacity, weekly production must be 20 one-person, 220 two-person, and 100 four-person canoes.

Here's a step-by-step solution for the entire problem:

a. Assign the variables of the problem using x, y, and z:

• 
  `\( x = \)` number of one-person canoes

• 
  `\( y = \)` number of two-person canoes

• 
  `\( z = \)` number of four-person canoes

b. Fill out the following table from the information given: ( Image is given below)

c. Write the augmented matrix for the system of equations:

The augmented matrix for the system of linear equations based on the table is:

`\[\begin{bmatrix}0.5 & 1 & 1.5 & 380 \\0.6 & 0.9 & 1.2 & 330 \\0.2 & 0.3 & 0.5 & 120 \\\end{bmatrix}\]`

d. Solve the system of equations using Gauss-Jordan Elimination:

The reduced row echelon form (RREF) of the augmented matrix is:

`\[\begin{bmatrix}1 & 0 & 0 & 20 \\0 & 1 & 0 & 220 \\0 & 0 & 1 & 100 \\\end{bmatrix}\]`

This matrix corresponds to the following equations:

• 
  `\( 1x + 0y + 0z = 20 \)`

• 
  `\( 0x + 1y + 0z = 220 \)`

• 
  `\( 0x + 0y + 1z = 100 \)`

Which directly gives us the values of
`\( x \)`,
`\( y \)`, and
`\( z \)`.

e. Using a complete sentence, how many canoes of each type must be produced each week for the plant to operate at full capacity?

To operate at full capacity, the plant must produce 20 one-person canoes, 220 two-person canoes, and 100 four-person canoes each week.

The complete question and the table is given below: