A construction firm employs two levels of title installers: craftsmen and apprentices. Craftsmen install 500 sq feet of specialty tile, 100 square feet of plain tile, and 100 linear feet of trim in one day. Apprentice installs 100 sq feet of specialty tile, 200 sq feet of plain title and 100 linear feet of trim in one day. How many are required to install 2000 sq feet of specialty title, 1600 sq feet of plain tile and 100 linear feet of trim if x = craftsmen and y=apprentices. Were craftsmen may $200 per day and apprentices make $120 per day.
Graph the minimum and maximum.
To find the number of craftsmen and apprentices needed to install the specified amount of tile, we can set up a system of equations.
Let x be the number of craftsmen and y be the number of apprentices.
The amount of specialty tile installed by craftsmen is 500 sq feet per day, so the total amount installed by x craftsmen is 500x sq feet per day.
The amount of specialty tile installed by apprentices is 100 sq feet per day, so the total amount installed by y apprentices is 100y sq feet per day.
Similarly, for plain tile:
Craftsmen install 100 sq feet per day => Total installed by x craftsmen = 100x sq feet per day.
Apprentices install 200 sq feet per day => Total installed by y apprentices = 200y sq feet per day.
For trim:
Craftsmen install 100 linear feet per day => Total installed by x craftsmen = 100x linear feet per day.
Apprentices install 100 linear feet per day => Total installed by y apprentices = 100y linear feet per day.
Now, let's set up the equations based on the given information:
500x = 2000 (specialty tile)
100x = 1600 (plain tile)
100x = 100 (trim)
Simplifying each equation, we get:
x = 4 (specialty tile)
x = 16 (plain tile)
x = 1 (trim)
Now, let's find the number of apprentices needed:
100y = 2000 (specialty tile)
200y = 1600 (plain tile)
100y = 100 (trim)
Simplifying each equation, we get:
y = 20 (specialty tile)
y = 8 (plain tile)
y = 1 (trim)
So, to install 2000 sq feet of specialty tile, 1600 sq feet of plain tile, and 100 linear feet of trim, we would need 4 craftsmen and 20 apprentices.
To graph the minimum and maximum, we need more information about the constraints or limitations of the problem. Without that information, we cannot accurately determine the minimum and maximum number of craftsmen and apprentices required.
To determine the number of craftsmen and apprentices required to install the given amounts of specialty tile, plain tile, and trim, we need to set up a system of equations.
Let's define the variables:
x = number of craftsmen
y = number of apprentices
Based on the given information, the installation rates for the craftsmen and apprentices are as follows:
For craftsmen:
Specialty tile: 500 sq ft/day
Plain tile: 100 sq ft/day
Trim: 100 linear ft/day
For apprentices:
Specialty tile: 100 sq ft/day
Plain tile: 200 sq ft/day
Trim: 100 linear ft/day
Therefore, the system of equations is:
500x + 100y = 2000 (eq. 1)
100x + 200y = 1600 (eq. 2)
100x + 100y = 100 (eq. 3)
To solve this system of equations, we can use various methods such as substitution or elimination. Let's use the elimination method:
Multiply eq. 2 by 5 to eliminate the x term:
500x + 1000y = 8000 (eq. 4)
Subtract eq. 1 from eq. 4 to eliminate the x term:
500x + 1000y - 500x - 100y = 2000 - 8000
900y = -6000
Divide both sides by 900:
y = -6000/900
y = -20/3
Since we cannot have a negative number of apprentices, we discard this solution.
Therefore, there is no feasible number of craftsmen and apprentices that can install the given amounts of tile and trim.
As for graphing the minimum and maximum, since there is no solution to the system of equations, there are no minimum and maximum points to graph.
To solve this problem, we can set up a system of equations based on the given information.
Let's denote the number of craftsmen as x and the number of apprentices as y.
From the information given, we can determine the following equations:
1. Craftsmen install 500 sq feet of specialty tile per day: x * 500 = 2000
2. Craftsmen install 100 sq feet of plain tile per day: x * 100 = 1600
3. Craftsmen install 100 linear feet of trim per day: x * 100 = 100
4. Apprentices install 100 sq feet of specialty tile per day: y * 100 = 2000
5. Apprentices install 200 sq feet of plain tile per day: y * 200 = 1600
6. Apprentices install 100 linear feet of trim per day: y * 100 = 100
Now, we have a system of equations:
x * 500 = 2000
x * 100 = 1600
x * 100 = 100
y * 100 = 2000
y * 200 = 1600
y * 100 = 100
To simplify the equations, we can divide each equation by the respective coefficient:
7. x = 4
8. x = 16
9. x = 1
10. y = 20
11. y = 8
12. y = 1
Now we have simplified equations:
x = 4
x = 16
x = 1
y = 20
y = 8
y = 1
To determine the minimum and maximum values, we can consider the constraints of the problem.
The minimum number of craftsmen required would be 1, as there must be at least one craftsman for any installation job.
The minimum number of apprentices required would also be 1, as there must be at least one apprentice for any installation job.
The maximum number of craftsmen would be determined by dividing the total area of specialty tile by the number of square feet installed by a craftsman per day:
Max craftsmen = 2000 (total sq ft of specialty tile) / 500 (sq ft installed by a craftsman per day) = 4
The maximum number of apprentices would be determined by dividing the total area of specialty tile by the number of square feet installed by an apprentice per day:
Max apprentices = 2000 (total sq ft of specialty tile) / 100 (sq ft installed by an apprentice per day) = 20
Graphing this situation on a graph would result in a straight line connecting the points (1, 1) and (4, 20), as these represent the minimum and maximum values respectively, for craftsmen and apprentices.
Please note that the salary information provided in the question is not relevant to finding the minimum and maximum values, but it is useful in determining the cost of labor for each scenario.