Craftsmen install 500 sq ft of ceramic tile and 100 sq ft of vinyl tile in one day. An apprentice installs 100 sq ft of ceramic tile and 200 sq ft of vinyl tile in one day. The firm has a job that requires 2000 sq ft of ceramic tile and 1600 sq ft of vinyl tile. Tessa pays craftsmen $200 per day and apprentices $120 per day. How many Craftsman and apprentices should she employ to minimize costs?

Question

Craftsmen install 500 sq ft of ceramic tile and 100 sq ft of vinyl tile in one day. An apprentice installs 100 sq ft of ceramic tile and 200 sq ft of vinyl tile in one day. The firm has a job that requires 2000 sq ft of ceramic tile and 1600 sq ft of vinyl tile. Tessa pays craftsmen $200 per day and apprentices $120 per day.  How many Craftsman and apprentices should she employ to minimize costs?

Reiny · Answer

number of craftmen --- x
number of apprentices --y

500x + 100y ≥ 2000 ---> 5x + y ≥ 20 for ceramic
100x + 200y ≥ 1600 ---> x + 2y ≥ 16 for vinyl

cost = 200x + 120y

sketch the first two relations in x and y
we need the closest integer-point to their intersection point of (2.67 , 6.67)
which would be (3,7)

So 3 craftsmen and 7 apprentices would do the job and be cheapest.

check:
with (3,8) --> 1500+800 or 2300 ceramic, job done
---------> 300+1600 or 1900 vinyl, job done
cost = 200(3) + 120(8) = $1560

with (3,7) --> 1500+700= 2200 ceramics, job done
----------> 300+ 1400= 1700 vinyl, job done
cost = 200(3) + 120(7) = $1440