A local entrepreneur is starting a moving company in the area and needs a fleet of trucks. They know they will be able to purchase at least 14 trucks, giving a total load capacity of at least 36 tons. One truck brand will hold 2 tons of cargo and costs $12000. A second truck brand will hold 3 tons of cargo and costs $22000.
Write the system of linear inequalities to represent this situation.
Sketch the feasible region.
Write the optimization equation.
Determine the optimal combination of vehicles to minimize the cost. Show all work.
Let's denote the number of trucks from the first brand as X and the number of trucks from the second brand as Y.
1. The entrepreneur knows they will be able to purchase at least 14 trucks, so the first inequality is:
X + Y ≥ 14
2. The total load capacity needs to be at least 36 tons, so the second inequality is:
2X + 3Y ≥ 36
Next, let's sketch the feasible region.
The feasible region represents all the possible combinations of trucks that satisfy the given constraints. To sketch it, we need to represent each inequality as a line on a graph and shade the region that satisfies both inequalities.
For the first inequality X + Y ≥ 14:
- If X = 0, then Y ≥ 14.
- If Y = 0, then X ≥ 14.
Drawing a line for X = 0 (the Y-axis) and a line for Y = 0 (the X-axis), they intersect at the point (14, 0) on the graph. The feasible region is to the right and above this point.
For the second inequality 2X + 3Y ≥ 36:
- If X = 0, then Y ≥ 12.
- If Y = 0, then X ≥ 18.
Drawing a line for X = 0 and a line for Y = 0, they intersect at the point (18, 0) on the graph. The feasible region is to the right and above this point.
Shade the region that satisfies both inequalities, which is the overlapping region to the right and above the points (14, 0) and (18, 0).
Now, let's write the optimization equation and determine the optimal combination of vehicles to minimize the cost.
The cost equation is given by the total cost of the trucks, which is:
Cost = 12000X + 22000Y
To minimize the cost, we need to find the combination of X and Y that satisfies the constraints and gives the lowest cost.
Now, we need to examine the corners of the feasible region (the points where the lines intersect) to see which combination of X and Y gives the minimum cost.
The corners of the feasible region are:
A: (14, 0)
B: (18, 0)
C: (27, 9)
D: (14, 12)
To find the cost at each corner, substitute the X and Y values into the cost equation.
Cost at A: 12000(14) + 22000(0) = $168,000
Cost at B: 12000(18) + 22000(0) = $216,000
Cost at C: 12000(27) + 22000(9) = $444,000
Cost at D: 12000(14) + 22000(12) = $396,000
The optimal combination of vehicles to minimize the cost is at point A: (14, 0) with a cost of $168,000.