Algebra 2 Word Problem

Question

A construction firm employs two levels of tile installers: craftsmen and apprentices. Craftsmen install 500 square feet of speciality tile, 100 square feet of plain tile, and 100 linear feet of trim in one day. An apprentice installs 100 square feet of speciality tile, 200 square feet of plain tile, and 100 linear feet of trim in one day. The firm has a one-day job that requires 2000 square feet of speciality tile, 1600 square feet of plain tile, and 1200 linear feet of trim.

Let x represent the number of craftsmen, and let y represent the number of apprentices. Write a system of 3 equations to represent the construction firms' situation with this job.

Answer 1

In this problem we can break down the company's productivity into the three abilities of craftsman and apprentices.

 

a=500x+100y amount of specialty tile laid per day

b=100x+200y amount of plain tile laid per day

c=100x+100y amount of trim laid per day

 

To solve the firm's manning problem you can either make a 5 column chart with 'x' in one column, 'y' in column two, and use the 3rd, 4th and 5th columns for the totals of each equation.

Substitute values for x and y until you come up with a reasonable solution.

 

x          y          a          b          c

1          1          600      300      200

2          1          1100    400      300

3          1          1600    500      400

...

0         12         1200    2400    1200

...

5         7           3200    1900    1200

 

At this point you can see that creating a a table can take forever.  I highly recommend using a graph with the three equations.  In this instance you know that the firm has specific requirements that they want meet or exceed.  

 

Graphing the three equations 

2000≤500x+100y (y≤-5x+20)

1600≤100x+200y (y≤-x+16)

1200≤100x+100x (y≤-.5x+6)

 

Desmos Graphing Calculaor is a great tool https://www.desmos.com/calculator

This will result in a region where the firm can satisfy its requirements.  

 

Just from looking at the graph you'll see that the following values would be suitable for the company.

x         y          a          b          c

0         20        2000    4000     2000

1         15

2         10        2000    2200     1200

3         9

4         8

5         7

6         6

7         5

8         4          4400    1600     1200

10       3

12       2

14       1

16       0          8000    1600     1600

 

At this point a company can go one step further.  What if they only have 5 craftsman? Or what if they want to find the most cost effective solution.  If craftsman gets paid $400.00 and the apprentice only gets paid $200.00, what is the cheapest solution if you have to have at least one craftsman on site for every three apprentices?