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Question 1144750: A tent designer is working on a new tent. The tent will be made from black fabric, which costs $7 per yard, and green fabric, which costs $6 per yard. The designer will need at least 4 yards of black fabric, at least 6 yards of green fabric, and at least 12 yards of fabric overall. The total cost of the fabric used for the tent can be no more than $135. Use this information for Items 1-3.
1. Let x represent the number of yards of black fabric and y represent the number of yards of green fabric.
Write inequalities that model the four constraints in this situation.
2. Graph the constraints on graph paper using a straight edge to draw the axes and lines. Be sure to label each
axis, the units, and lines. You should have work shown to validate your shading.
3. Identify two ordered pairs that are solutions. Explain the meaning of each ordered pairs in a complete
sentence.
4. What is the greatest amount of green fabric the designer can use if all the constraints are met? Explain your
answer in a complete sentence.
5. What is the least amount of black fabric the designer can use if all the constraints are met. Explain your
answer in a complete sentence.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x represents the number of yards of black fabric.
y represents the number of yards of green fabric.
your constraints are:
x >= 4
y >= 6
x + y >= 12
7x + 6y <= 135
you would graph the equality portion of these inequalities.
specifically, you would graph:
x = 4
y = 6
x + y = 12
7x + 6y = 135
you would then shade the area of the graph that satisfies the inequalities.
your graph would look like this.
your region of feasibility would be the area......
to the right of x = 4
above y = 6
above x + y = 12
below 7x + 6y = 135
this corresponds to your constraint inequalities shown above.
the corner points of the feasible region are the points where your maximum or minimum values for the objective function would be.
these corner points are where the solution to your problem lies.
this problem does not have an objective function, so you would not do an evaluation at any of the corner points.
one objective function might be to minimize your costs.
in that case, your objective function would be cost = 7x + 6y.
you would evaluate your objective function at each of the corner points and then pick the corner point that has the least cost.
the answer to the questions that you do have are shown below.
1. Let x represent the number of yards of black fabric and y represent the number of yards of green fabric.
Write inequalities that model the four constraints in this situation.
done above.
2. Graph the constraints on graph paper using a straight edge to draw the axes and lines. Be sure to label each
axis, the units, and lines. You should have work shown to validate your shading.
done above.
3. Identify two ordered pairs that are solutions. Explain the meaning of each ordered pairs in a complete
sentence.
(4,8) is one solution.
it satisfies the constraints as follows:
x >= 4
y >= 6
x + y = 4 + 8 = 12 >= 12
7x + 6y = 7 * 4 + 6 * 8 = 28 + 48 = 76 <= 135
(14.143,6) is another solution.
it satisfies the constraints as follows:
x >= 4
y >= 6
x + y = 14.143 + 6 = 20.143 >= 12
7x + 6y = 7 * 14.143 + 6 * 6 = 135 <= 135 (true if you take away the rounding to 3 decimal places that the graph shows - un-rounded value would be approximately 14.142857 as found on my TI-84 Plus).
4. What is the greatest amount of green fabric the designer can use if all the constraints are met? Explain your answer in a complete sentence.
the greatest amount of green fabric would be at the point (4,17.833).
that where the value of y is the greatest.
at that point, the constraint that limits the value of y to be greater is the constraint of 7x + 6y <= 135.
5. What is the least amount of black fabric the designer can use if all the constraints are met. Explain your answer in a complete sentence.
the least amount of black fabric would be at the points (4,8) and (4,17.833).
that's where the value of x is the smallest.
that satisfies the constraint that x >= 4.
(4,17.833) also satisfies the constraint that 7x + 6y <= 135.
(4,8) also satisfies the constraint that x + y >= 12.
you basically satisfies all the constraint inequalities at each of the corner points.
all of the corner points should satisfy all of the constraints.
in this graph, they do.
if any of the constraints are not satisfied at a particular corner point, then that point is invalid.
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