Problem 44 - College Algebra with Corequisite Support

Question

To reach escape velocity, a rocket must travel at the rate of \(2.2 \times 10^{6} \mathrm{ft} / \mathrm{min}\). Rewrite the rate in standard notation.

Step-by-Step Solution

Answer

The rate in standard notation is 22,000,000 ft/min.

Step 1: Understand Scientific Notation

The given rate is in scientific notation: \(2.2 \times 10^{6}\). This means 2.2 should be multiplied by 10 raised to the power of 6.

Step 2: Calculate the Power of Ten

The power of 10 to the sixth, \(10^6\), represents multiplying 10 by itself 6 times, or \(10,000,000\).

Step 3: Expand the Scientific Notation

Multiply 2.2 by the value above: \[2.2 \times 10^6 = 2.2 \times 10,000,000 = 22,000,000 \].

Step 4: Write in Standard Notation

The standard notation for the given rate is \(22,000,000\) feet per minute.

Key Concepts

Standard Notation

Standard notation is the way of writing numbers in the usual numerical form—using digits. This means there are no powers or exponents involved. When expressing a number like the rocket's rate of travel in the exercise, converting from scientific to standard notation makes it easier to read and understand.

Scientific notation was used initially to express a large number more compactly; however, students must also recognize its equivalent in standard notation. Modeling proficiency in translating between these notations can be key in many scientific and engineering fields.

• Standard notation: Regular way to write numbers, using full decimal expression.
• Scientific notation: A method to represent very large or small numbers using powers of ten.

In the case of the exercise, the standard notation for the escape velocity given as \(2.2 \times 10^{6}\) is \(22,000,000\), making it a larger yet more intuitive form for direct reading.

Power of Ten

The power of ten is a fundamental concept in mathematics, used to express large or small numbers in powers which involve the number 10 raised to an exponent. It's crucial when learning about scientific notation. The number 10 is repeatedly multiplied by itself by a number of times determined by the exponent, which shows how far the decimal point has "moved."

When dealing with \(10^6\), we consider multiplying 10 by itself six times:

• \(10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000\).

This practice translates scientific notation directly into a numerical equivalent:

• The number \(10^6\) equates to 1,000,000, a useful translation when converting quantities into standard notation.

By understanding this, students can better navigate tasks requiring adjustments of numerical scales, whether magnifying or diminishing values.

Multiplication

Multiplication is a basic arithmetic operation, vital for translating scientific notation into standard form. Within this context, multiplying involves taking the leading digit in the scientific notation and scaling it by the power of ten.

In mathematical terms, you multiply the coefficient (2.2 in this instance) by the enormous number the power of ten represents (\(10^6\) here is 1,000,000). Explicitly working through this:

• \(2.2 \times 1,000,000 = 22,000,000\).

This straightforward multiplication requires students to recognize how the decimal shift correlates to the exponent on the 10.

Students should also appreciate how multiplication can help streamline their handling of big numbers, breaking them into simpler components and recomposing them into standard notation.