Question

A man finds that he has a mass of 100.6 \(\mathrm{kg} .\) He goes on a diet, and several months later he finds that he has a mass of 96.4 \(\mathrm{kg}\) . Express each number in scientific notation, and calculate the number of kilograms the man has lost by dieting.

Short answer

The man has lost 4.2 kg.

Step-by-step solution

Convert Initial Mass to Scientific Notation

The initial mass of the man is given as 100.6 kg. To convert this to scientific notation, move the decimal point so that there is one non-zero digit to its left. Hence, 100.6 kg can be written as \(1.006 \times 10^2 \text{ kg}\).

Convert Final Mass to Scientific Notation

The final mass of the man is given as 96.4 kg. Similarly, convert this to scientific notation by moving the decimal point so that there is one non-zero digit to its left. Thus, 96.4 kg can be rewritten as \(9.64 \times 10^1 \text{ kg}\).

Calculate the Mass Lost

Subtract the final mass from the initial mass to find out how much weight the man has lost: \(100.6 \text{ kg} - 96.4 \text{ kg}\).\ Every kilogram from scientific notation variables should be considered in standard form:\( 100.6 - 96.4 = 4.2 \text{ kg} \). The man has lost 4.2 kg.

Key concepts

Mass Conversion

Mass conversion is a fundamental concept in science, where we often need to express large or small masses in a more manageable form. Here, we move decimals to convert masses to scientific notation. For example, to express a mass of 100.6 kg in scientific notation, we shift the decimal place to get a number between 1 and 10. We then multiply this number by a power of ten. Thus, 100.6 kg becomes 1.006 × 10^2 kg. Similarly, if a mass is 96.4 kg, we convert it by moving the decimal point, resulting in 9.64 × 10^1 kg.

Weight Loss Calculation

Calculating weight loss is important in various scientific and health fields. To determine weight loss, we subtract the final mass from the initial mass. First, it's helpful to convert masses into a common format, such as scientific notation.

In our example, the initial mass (100.6 kg) converts to 1.006 × 10^2 kg, and the final mass (96.4 kg) converts to 9.64 × 10^1 kg. To find the weight lost:

• First convert both values back to standard form, which are 100.6 kg and 96.4 kg respectively.
• Subtract the final mass from the initial mass: 100.6 kg - 96.4 kg.
• The difference is 4.2 kg, indicating the man lost 4.2 kg.

This simple subtraction helps us understand how much weight was lost.

Scientific Notation Subtraction

Subtracting numbers in scientific notation is a common operation in scientific calculations. To do this, ensure the numbers have the same exponent. If not, convert them so they do. In the exercise, we were given masses in standard form and converted them to scientific notation.

Our initial mass was 1.006 × 10^2, and the final mass was 9.64 × 10^1.

• First, recognize that 9.64 × 10^1 can be rewritten as 0.964 × 10^2 to match the exponent of the initial mass.
• Then, perform the subtraction: 1.006 × 10^2 - 0.964 × 10^2.
• This equals (1.006 - 0.964) × 10^2, which simplifies to 0.042 × 10^2.
• Finally, convert it back to standard form to get 4.2 kg.

This method helps simplify subtraction when dealing with large or small numbers efficiently.