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?
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
To minimize costs, Tessa needs to determine the optimal number of craftsmen and apprentices to employ. Let's represent the number of craftsmen she employs as "C" and the number of apprentices as "A".
Given the productivity rates for craftsmen and apprentices, we can calculate the total number of days it would take to complete the job using the formula:
Total Days = Total Area / (Craftsman's Daily Area + Apprentice's Daily Area)
For ceramic tile:
Craftsman's Daily Area = 500 sq ft
Apprentice's Daily Area = 100 sq ft
Total Area of Ceramic Tile = 2000 sq ft
Total Days for Ceramic Tile = 2000 / (500C + 100A)
For the vinyl tile:
Craftsman's Daily Area = 100 sq ft
Apprentice's Daily Area = 200 sq ft
Total Area of Vinyl Tile = 1600 sq ft
Total Days for Vinyl Tile = 1600 / (100C + 200A)
The total cost can be calculated by multiplying the number of days by the daily wage for craftsmen and apprentices:
Total Cost = (Craftsman's Daily Wage * Total Days for Ceramic Tile * C) + (Apprentice's Daily Wage * Total Days for Vinyl Tile * A)
Craftsman's Daily Wage = $200
Apprentice's Daily Wage = $120
So, the total cost is:
Total Cost = (200 * (2000 / (500C + 100A)) * C) + (120 * (1600 / (100C + 200A)) * A)
To minimize costs, Tessa needs to minimize the total cost equation by finding the values of C and A that minimize it. This can be done by using optimization techniques such as differentiation or by using trial and error to test different values of C and A.