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?
To determine the number of craftsmen and apprentices Tessa should employ to minimize costs, we need to set up a system of equations based on the given information.
Let's assume the number of craftsmen to be x and the number of apprentices to be y.
Based on the given information, we can set up the following equations:
1) For ceramic tile installation:
Craftsmen: x craftsmen install 500 sq ft in one day
Apprentices: y apprentices install 100 sq ft in one day
So the total ceramic tile installation in one day is:
500x + 100y sq ft
2) For vinyl tile installation:
Craftsmen: x craftsmen install 100 sq ft in one day
Apprentices : y apprentices install 200 sq ft in one day
So the total vinyl tile installation in one day is:
100x + 200y sq ft
Given that the firm requires 2000 sq ft of ceramic tile and 1600 sq ft of vinyl tile, we can set up the following equations:
3) For ceramic tile:
500x + 100y = 2000
4) For vinyl tile:
100x + 200y = 1600
Now, we need to minimize the cost. The cost consists of the total payment to craftsmen and apprentices.
The cost for craftsmen is $200 per day, so the cost for x craftsmen in one day is 200x.
The cost for apprentices is $120 per day, so the cost for y apprentices in one day is 120y.
Now the total cost can be expressed as:
Cost = 200x + 120y
To solve this system of equations and find the values of x and y, we can use any method of solving linear equations, such as substitution or elimination.
Let's solve it using substitution:
From equation 3), we can rearrange it to get:
500x = 2000 - 100y
x = (2000 - 100y) / 500
Now substitute this value of x in equation 4):
100((2000 - 100y) / 500) + 200y = 1600
multiply through by 500 to clear the fraction:
100(2000 - 100y) + 1000y = 800000
Simplify and rearrange the equation:
200000 - 10000y + 1000y = 800000
-9000y = 800000 - 200000
-9000y = 600000
y = 600000 / -9000
y = -66.67
We can't have a fraction of an apprentice, so y should be a whole number. Let's round it to the nearest whole number, which will be -67.
Now substitute this rounded value of y back into the equation to find x:
500x + 100*(-67) = 2000
500x - 6700 = 2000
500x = 8700
x = 8700 / 500
x = 17.4
Again, x should be a whole number, so let's round it to the nearest whole number, which will be 17.
Therefore, Tessa should employ 17 craftsmen and -67 apprentices. Since we can't have a negative number of apprentices, we need to discard the negative value. Hence, Tessa should employ 17 craftsmen to minimize costs.