Choosing Appropriate Units with Scientific Notation

Choosing Appropriate Units with Scientific Notation exercise

• What is scientific notation?

  Hints

  Our knowledge of powers of ten helps us compare numbers in scientific notation by understanding that multiplying by 10 raises the number by one power, and dividing by 10 lowers the number by one power.

  For instance, comparing $4.5 \times 10^6$ and $7.2 \times 10^5$, we know that $4.5 \times 10^6$ is larger because it has a higher power of ten ($6$ compared to $5$).

  Solution

  Scientific notation is a way of writing numbers that are very large or very small using powers of ten.

  For example, $6.02 \times 10^{23}$ represents a very large number, and $3.2 \times 10^{-5}$ represents a very small number.

• Which unit of measurement is more appropriate for expressing the length?

  Hints

  Reflect on the standard units used for measuring distances and their relevance in everyday contexts.

  Which unit is commonly used for larger distances, especially when discussing geographic features like rivers?

  When dealing with large distances, using a consistent unit like miles allows for easier comparisons and calculations.

  It avoids the need for extensive conversions and simplifies data analysis.

  Solution

  Sarah's measurement is $2.02752 \times 10^{5}$ inches.

  Jack's measurement is $3.2$ miles.

  In this case, measuring in miles would be more appropriate for determining the length of the river.

• Compare these measurements and decide which one is easier to work with in calculations.

  Hints

  Think about the scale of the measurement.

  Consider whether the mass being measured is relatively large or small, and choose a unit of mass that makes sense in that context.

  Reflect on the precision required for the measurement.

  Consider whether using grams alone is sufficient, or if you need to consider smaller units like milligrams for more accurate measurements.

  Solution

  45 grams.

  If you need a more precise measurement or are dealing with smaller masses, grams would be a better option.

  For example, in scientific experiments or when measuring ingredients in cooking, grams are commonly used.

• Determine which unit of measurement is more practical for describing the pool's volume.

  Hints

  Consider the scale of the measurement.

  Think about which unit (cubic inches or cubic feet) would result in a more manageable and easy-to-understand number for describing the size of a swimming pool.

  Reflect on which unit (cubic inches or cubic feet) is commonly used for larger volumes like swimming pools in everyday conversations or real-world applications.

  Solution

  Using cubic feet is more practical for describing the pool's volume because it results in a smaller, more manageable number without the need for scientific notation.

  Additionally, cubic feet is a commonly used unit for measuring larger volumes like swimming pools, making it more familiar and understandable in everyday contexts.

• Convert a number in scientific notation to standard form.

  Hints

  Consider the power of $10$.

  The exponent $2$ indicates how many places to move the decimal point.

  Recall that $10^2$ = $100$.

  Solution

  $5 \times 10^2$ in standard form is $500$.

• Convert numbers in scientific notation to standard form.

  Hints

  Meters are a standard unit of length used in scientific contexts, especially for large distances like the distance from Earth to the Moon.

  Using meters provides a more appropriate and universally understood.

  Using grams provides a more precise and detailed measurement for a scientific experiment analysis.

  Using cubic feet instead of cubic inches would be more practical for measuring the volume of a swimming pool.

  Since swimming pools are typically large structures, using cubic feet provides a more manageable and understandable measurement.

  Additionally, working with cubic feet allows for easier visualization of the pool's size and simplifies calculations.

  Solution

  The distance from Earth to the Moon - $3.84\times 10^{5}$ meters

  The mass of a tennis ball - $7.2\times 10^{5}$ grams

  The volume of a swimming pool - $9.2\times 10^{4}$ cubic feet