Problem 27 - Elementary Algebra

Question

Find the total amount the company is paying for pollution controls and the penalty. Write the answer in place value notation. Last July, East Kentucky agreed to install pollution controls estimated to cost \(\$ 650\) million and to pay a \(\$ 750,000\) penalty to resolve violations of the new source review provisions of the Clean Air Act at the Dale facility and two other plants. (Source: www.yosemite.epa.gov, Sept. 20, 2007)

Step-by-Step Solution

Answer

The company is paying 650.75 million dollars, which is 600 + 50 + 0.75 million dollars.

Step 1 - Understand the Given Values

Identify the costs given in the problem. The pollution control costs are given as \(650 million, and the penalty cost is given as \)750,000.

Step 2 - Convert Costs to Same Units

Convert the penalty amount from dollars to millions of dollars to have uniform units.Penalty in millions: \[ \frac{750,000}{1,000,000} = 0.75 \text{ million dollars} \]

Step 3 - Add the Costs

Add the pollution control cost and the penalty (both in millions).Total cost: \[ 650 + 0.75 = 650.75 \text{ million dollars} \]

Step 4 - Convert to Place Value Notation

To express 650.75 in place value notation: 650.75 = 600 + 50 + 0.75

Key Concepts

Unit Conversion

Understanding unit conversion is crucial for solving many math problems, including this one. In the exercise, we needed to convert the penalty amount from dollars to millions of dollars.
This is done because the pollution control cost was already in millions, so converting both costs into the same unit makes addition straightforward.
To convert \$750,000\$$ into millions: Divide by 1,000,000, resulting in 0.75 million dollars since \(\frac{750,000}{1,000,000} = 0.75\).
Always ensure units match before performing operations like addition or subtraction.

Addition of Decimals

Adding decimals may seem intimidating, but it’s simple once you align the decimal points. In our problem, we needed to add \$650\$$ million and \$0.75\$$ million.
Ensure both numbers are written with the decimal points aligned one below the other:
\[\begin{array}{r} 650.00 \ + 0.75 \ \hline 650.75 \end{array}\].
Sum each column starting from the right.
When done correctly, this results in \$650.75\$$ million.

Place Value Notation

Place value notation helps to express numbers in a detailed and clear way, showing the value of each digit.
In the exercise, the result was \$650.75\$$ million. We can break this down using place value notation:
\ 650.75 = 600 + 50 + 0.75 \
Here:
- 600 captures the hundreds place
- 50 captures the tens place
- 0.75 covers the portion of the number less than one. This method makes it easier to understand and verify the calculation.