Problem 39 - Thinking Mathematically

Question

A building contractor is to dig a foundation 12 feet long, 9 feet wide, and 6 feet deep for a toll booth. The contractor pays \(\$ 85\) per load for trucks to remove the dirt. Each truck holds 6 cubic yards. What is the cost to the contractor to have all the dirt hauled away?

Step-by-Step Solution

Answer

The total cost to have all the dirt hauled away will be \$340.

Step 1: Calculate the volume of the foundation in cubic feet

The volume of the foundation is calculated by multiplying its length, width, and depth. Hence the volume \(V\) in cubic feet is given by the formula \(V = 12ft \times 9ft \times 6ft = 648 cubic feet\).

Step 2: Convert the volume from cubic feet to cubic yards

Conversion from cubic feet to cubic yards is performed according to the standard conversion rate, 1 cubic yard = 27 cubic feet. The total volume \(V_y\) in cubic yards is therefore \(V_y = 648 ft^3 / 27 = 24 cubic yards\).

Step 3: Calculate the number of truckloads required

The number of truckloads is obtained by dividing the total volume of dirt by the volume that each truck can carry. In this case, each truck can carry 6 cubic yards. Therefore, the number of trucks needed is \(n = V_y / 6 = 24 / 6 = 4\) trucks.

Step 4: Calculate the total cost

The total cost is calculated by multiplying the number of trucks by the cost per truck, i.e., \(cost = n \$85 = 4 \times \$85 = \$340\).

Key Concepts

Volume Calculation

One of the first steps in solving problems involving physical spaces is to determine the volume. Volume is a measure of the amount of space an object occupies. For the foundation of the toll booth, the volume is calculated using the formula for the volume of a rectangle: \[V = \, \text{length} \times \, \text{width} \times \, \text{depth} \] Plugging in the given dimensions of the foundation:

• Length: 12 feet
• Width: 9 feet
• Depth: 6 feet

The volume calculation becomes: \[V = 12 \, ext{ft} \times 9 \, \text{ft} \times 6 \, \text{ft} = 648 \, ext{cubic feet}\] This tells us the total space, in cubic feet, occupied by the imagined foundation.

Unit Conversion

When dealing with measurements, converting units is crucial for ensuring consistency, especially in large calculations. Here, the volume calculated in cubic feet needs to be converted to cubic yards, as the truck's capacity is given in cubic yards. Knowing the conversion factor helps.

Understanding the Conversion Factor

• 1 cubic yard is equivalent to 27 cubic feet.

To convert from cubic feet to cubic yards, divide the volume in cubic feet by 27:\[V_y = \frac{648 \, \text{cubic feet}}{27} = 24 \, \text{cubic yards}\] Converting units in this way ensures that all subsequent calculations, especially those involving costs, are based on the correct measure of volume.

Cost Calculation

This section will guide you through the calculation of costs, which is a primary step in decision-making for projects. After knowing the total cubic yards of dirt, you can estimate the cost of hauling it away based on the truckload capacity.

Calculating Number of Truckloads

To determine the number of truckloads needed, divide the total volume in cubic yards by the volume a single truck can carry:\[n = \frac{V_y}{6 \, \text{cubic yards per truck}} = \frac{24}{6} = 4 \, \text{trucks}\] Each truckload costs \(\\( 85\). Therefore, the total cost becomes:\[\text{Total Cost} = n \times \\) 85 = 4 \times \\( 85 = \\) 340\] Understanding this cost calculation is crucial for budgeting and planning in construction management.

Problem-Solving Skills

Enhancing problem-solving skills involves understanding the complete procedure of dealing with real-life math problems. This involves an ability to break down a problem into manageable steps and tackling each effectively.

Structured Problem-Solving

• Clearly define the problem: Here it was about removing the dirt economically.
• Identify necessary calculations: Determine the volume, convert units, calculate number of truckloads, and cost.
• Apply arithmetic and algebraic principles step-by-step.
• Check each part of the process to ensure accuracy: Double-check calculations at each step.

Developing these skills can empower students to tackle more complex problems with confidence, knowing they can apply similar principles.