Problem 79 - Beginning and Intermediate Algebra

Question

Write the answer using scientific notation. It is estimated that there were 51.2 billion pieces of trash on 76 million mi of U.S. roadways in a recent year (Source: Keep America Beautiful). On average, how many pieces of trash were on each mile of roadway? (THE IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

On average, there were approximately \(6.73684 \times 10^2\) or 673.684 pieces of trash on each mile of roadway.

Step 1: We have 51.2 billion trash pieces and 76 million miles of roadway. We need to convert them into scientific notation. 51.2 billion = 5.12 x 10^10 ; 76 million = 7.6 x 10^7 ; #Step 2: Calculate the average trash pieces per mile#

To find the average trash pieces per mile, we will divide the total trash pieces by the total miles of roadways. Average = (Total Trash Pieces) / (Total Miles) ; #Step 3: Plug the numbers in the equation and solve#

Step 2: Now, we will plug the numbers in the above equation and solve for the average. Average = (5.12 x 10^10) / (7.6 x 10^7) ; #Step 4: Simplify the equation and write the answer in scientific notation#

Simplify the above equation by dividing the non-exponential parts first and then the exponential parts. Average = (5.12/7.6) x (10^10÷10^7) ; Average = 0.673684 x 10^3 ; Now, to write the final answer in scientific notation, move the decimal point three places to the right and adjust the exponent accordingly. Average = 6.73684 x 10^2 ; So, on average, there were approximately 6.73684 x 10^2 or 673.684 pieces of trash on each mile of roadway.

Key Concepts

Average Calculation

When calculating an average, it involves determining the central value of a set of numbers. In this case, we're trying to find out how many pieces of trash are present per mile of roadway. The formula for calculating an average is fairly straightforward, as it involves dividing the sum total of all items in a set by the number of items in that set.

• Total Number of Pieces: 51.2 billion
• Total Miles: 76 million
• Formula: Average = \( \frac{\text{Total Trash Pieces}}{\text{Total Miles}} \)

To find the average number of trash pieces per mile, you just need to put the above values into this formula and then simplify.

Division of Exponents

Dividing numbers associated with powers or exponents can be simplified by applying the rules of exponents. When dividing terms with the same base, you subtract the exponents. Here, we have two numbers written in scientific notation: \( 5.12 \times 10^{10} \) and \( 7.6 \times 10^7 \). To perform the division, follow these steps:

• Divide the coefficients: 5.12 ÷ 7.6
• Subtract the exponents: \( 10^{10} ÷ 10^7 = 10^{10-7} = 10^3 \)

Thus, the division of these two numbers in scientific notation becomes \( \frac{5.12}{7.6} \times 10^3 \). This results in the simplified value that needs further adjustment to match scientific notation norms.

Conversion to Scientific Notation

Scientific notation is a compact way of writing very large or very small numbers. It uses powers of 10 to express how many zeros would be in a full representation of the number. After the division of our numbers, we obtain \( 0.673684 \times 10^3 \).To convert this to proper scientific notation:

• Adjust the decimal to the right so the decimal is after the first non-zero digit: 6.73684
• The decimal was shifted 3 places to the right, reducing the power by 3: \( 10^3 \) becomes \( 10^2 \)

Thus, the final answer in scientific notation is \( 6.73684 \times 10^2 \), indicating that approximately 673.684 pieces of trash exist per mile on average.

Algebraic Simplification

Algebraic simplification involves breaking down complex arithmetic expressions into simpler or more manageable forms. In the context of this problem, simplification was necessary both in dividing the coefficients \( \frac{5.12}{7.6} \) and handling the exponents \( 10^{10} \) and \( 10^7 \).

• Simplify Divisions: Compute \( \frac{5.12}{7.6} \) to get \( 0.673684 \).
• Simplify Exponents: Subtract the powers to simplify \( 10^{10} \) and \( 10^7 \) to \( 10^3 \).
• Combine Results: Multiply the simplified coefficient by the simplified exponent part: \( 0.673684 \times 10^3 \).

Algebraic simplification helps to condense calculations and prepares them for conversion into scientific notation or further manipulation, making it essential in solving scientific and mathematical problems effectively.