Question

For construction of a civil engineering facility, a contractor has found natural reserves of sand and gravel at Bloomingdale and Valley Springs where he may purchase such material. The unit cost, including delivery from Bloomingdale and Valley Springs, is $5 and $7, respectively. After the material is brought to the site, it is mixed thoroughly and uniformly, and the contract specifications state that the mix should contain a minimum of 30% sand. A total volume of 100,000 m3 of mixed material is needed for the project. The Bloomingdale Pit contains 25% sand and the Valley Springs Pit contains 50% sand. As the new young construction engineer on the project, you are asked to determine how much material should be taken from each pit in order to minimize the cost of material.

Answer 1

To minimize the cost of material for the project, we need to set up a system of equations based on the given information and constraints. Solving this system will give us the optimal solution. The optimal solution is to take 93,333 m3 of material from Bloomingdale and 6,667 m3 of material from Valley Springs.

Step-by-step explanation:

To determine the amount of material that should be taken from each pit in order to minimize the cost, we need to set up a system of equations based on the given information and constraints. Let's assume that x is the volume of material (in cubic meters) taken from the Bloomingdale Pit and y is the volume of material (in cubic meters) taken from the Valley Springs Pit.

According to the contract specifications, the mixed material should contain a minimum of 30% sand. This means that the amount of sand in the mix should be at least 30% of the total volume. We can express this constraint as:

0.25x + 0.5y >= 0.3 * (x + y)

We also know that the total volume of mixed material needed is 100,000 m3, so:

x + y = 100,000

We can now solve these equations to find the values of x and y that minimize the cost.

Since the cost of material from Bloomingdale is $5 per unit volume, the cost of material from Bloomingdale is 5x. Likewise, the cost of material from Valley Springs is $7 per unit volume, so the cost of material from Valley Springs is 7y. The total cost can be expressed as:

Total Cost = 5x + 7y

To minimize the cost, we can use the method of substitution to solve the system of equations. First, solve the second equation for x:

x = 100,000 - y

Substitute this value of x into the first equation:

0.25(100,000 - y) + 0.5y >= 0.3 * (100,000)

Solve this inequality for y:

25,000 - 0.25y + 0.5y >= 30,000

0.25y + 0.5y <= 30,000 - 25,000

0.75y <= 5,000

y <= 6,667

Since y represents the volume of material from Valley Springs, it must be less than or equal to 6,667 m3. Substitute this value of y back into the second equation to find x:

x = 100,000 - 6,667

x = 93,333

So, the optimal solution is to take 93,333 m3 of material from Bloomingdale and 6,667 m3 of material from Valley Springs to minimize the cost of material for the project.

Answer 2

The contractor should purchase 40,000 m3 of material from Bloomingdale and 60,000 m3 of material from Valley Springs to minimize the total cost of material while meeting the contract specifications.

Let x be the amount of material purchased from Bloomingdale and y be the amount of material purchased from Valley Springs.

The objective function to minimize the total cost is:

Minimize: 5x + 7y

The constraint that the mix should contain at least 30% sand can be expressed as:

0.25x + 0.5y ≥ 0.3(x + y)

Simplifying this inequality, we get:

0.25x - 0.2y ≥ 0

The constraint that the total volume of mixed material should be 100,000 m3 can be expressed as:

x + y = 100,000

Converting this constraint to a form suitable for linear programming, we get:

x + y ≤ 100,000

The non-negativity constraints are:

x ≥ 0

y ≥ 0

Using the simplex method, we find the optimal solution is:

x = 40,000 m3

y = 60,000 m3

Therefore, the contractor should purchase 40,000 m3 of material from Bloomingdale and 60,000 m3 of material from Valley Springs to minimize the total cost of material while meeting the contract specifications.