Problem 82 - Beginning and Intermediate Algebra

Question

The 17.6 -mi long Chesapeake Bay Bridge-Tunnel was completed in \(1964 .\) Construction costs were \(\$ 210\) million. Find the average cost per mile. (THE IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

The average cost per mile for the Chesapeake Bay Bridge-Tunnel was approximately \(\$11.93\) million per mile.

Step 1: List given information

We are given the following information: - Total length of the Chesapeake Bay Bridge-Tunnel: \(17.6\) miles. - Total cost: \(\$210\) million.

Step 2: Set up the division

To find the average cost per mile, we will divide the total cost by the total length: \[\text{Average Cost per Mile} = \frac{\text{Total Cost}}{\text{Total Length}}\]

Step 3: Plug in given values

Substitute the given values into the formula: \[\text{Average Cost per Mile} = \frac{\$210 \text{ million}}{17.6\text{ miles}}\]

Step 4: Perform the division

Now, divide \(\$210\) million by \(17.6\text{ miles}\) to find the average cost per mile: \[\text{Average Cost per Mile} = \frac{\$210 \text{ million}}{17.6\text{ miles}} =\$11.93\text{ million per mile}\]

Step 5: Write the conclusion

The average cost per mile for the Chesapeake Bay Bridge-Tunnel was approximately \(\$11.93\) million per mile.

Key Concepts

Division in Mathematics

Division is one of the basic arithmetic operations, alongside addition, subtraction, and multiplication. It involves distributing a quantity into equal parts. Specifically, division indicates how many times one number, known as the divisor, fits into another number, known as the dividend.

In the context of our problem involving the Chesapeake Bay Bridge-Tunnel, we are tasked with using division to find out how much each mile of the tunnel costs. We do this by dividing the total cost of the construction by the total length of the bridge. This helps us determine the average cost per mile. For mathematical clarity, the process looks like this:

• Dividend: Total Cost (\(210 \text{ million}\) dollars)
• Divisor: Total Length (\(17.6\text{ miles}\))
• Quotient: Average Cost per Mile

When viewed as a simple formula: \[ \text{Average Cost per Mile} = \frac{\text{Total Cost}}{\text{Total Length}} \]
Breaking down tasks using division can simplify complex problems, providing a clearer perspective on how resources are distributed.

Word Problems in Algebra

Word problems are a fundamental element of algebra where real-world situations are translated into mathematical expressions or equations. The aim is to find the unknown by carefully analyzing the information provided in the problem statement.

In our example with the Chesapeake Bay Bridge-Tunnel, we start by identifying the key pieces of information:

• "Total cost was \(\$210\) million"
• "The bridge is \(17.6\) miles long"

The challenge is to convert this information into a mathematical model that allows us to determine the average cost per mile. In this way, word problems teach us how to interpret and process information, equipping us with problem-solving skills that are applicable beyond math.

Once the mathematical expression is set up, the method of solving becomes straightforward, but it all begins with understanding and accurately interpreting the problem.

Unit Rate Calculation

The concept of unit rate helps us determine how one quantity varies with another. It is essentially the cost or value of a single unit of measurement. Calculating the unit rate is especially valuable for determining efficiency, costs, or even work speed.

In our scenario of the Chesapeake Bay Bridge-Tunnel, the unit rate is the cost for one mile of the bridge. By dividing the total cost by the number of miles, we find the average cost per one mile, which is a specific example of a unit rate calculation.

The steps for calculating a unit rate often include:

• Identifying the total quantity (e.g., total cost)
• Identifying the total number of units (e.g., number of miles)
• Calculating the unit rate by division:\[ \text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Total Number of Units}} \]

By focusing on the unit rate, we gain insight into how efficiently resources are being utilized or distributed across the individual units.