Problem 44 - College Algebra

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

2,200,000 ft/min

Step 1: Understand the Problem

We are given a velocity in scientific notation, specifically \( 2.2 \times 10^6 \text{ ft/min} \), and we need to convert this into standard notation.

Step 2: Interpret Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small conveniently. Here, \( 2.2 \times 10^6 \) means 2.2 multiplied by 1,000,000.

Step 3: Perform the Multiplication

Multiply 2.2 by 1,000,000 to convert the velocity to standard form. Calculate \( 2.2 \times 10^6 = 2.2 \times 1,000,000 \).

Step 4: Write in Standard Notation

Calculate \( 2.2 \times 1,000,000 = 2,200,000 \). Thus, the rate of the rocket in standard form is \( 2,200,000 \text{ ft/min} \).

Key Concepts

Standard Notation

Standard notation is the usual way of writing numbers that doesn’t involve powers of ten. It allows us to express numbers as they are used in everyday situations. When converting from scientific notation to standard notation, we essentially expand the number so it's easier to read and comprehend. In this case, the number given in scientific notation is \(2.2 \times 10^6\). To write this in standard notation, we multiply 2.2 by 1,000,000. This is because the exponent "6" tells us to move the decimal point 6 places to the right. So, \(2.2\) becomes \(2,200,000\).

• If the exponent is positive, move the decimal right.
• If the exponent is negative, move the decimal left.

Standard notation is extremely useful when dealing with real-world measurements, such as distances or speeds, where scientific precision is less emphasized.

Mathematics

Mathematics is a vast field, integral to almost every area of science and daily life. It provides the tools needed to translate, solve, and communicate complex problems. Scientific notation is a mathematical concept used to simplify the handling of very large or small numbers. This notation relies on powers of ten to express quantities compactly. It's particularly useful in fields like astronomy or physics, where extreme values are common. Basic understanding of how to convert between different notations is a mathematical skill that enhances problem-solving. Understanding and using mathematical principles, such as those in scientific notation, helps with:

• Effective problem-solving strategies.
• Accurate communication of numerical information.
• Efficient data analysis across multiple fields.

Velocity Conversion

Velocity conversion is the process of converting speed measurements from one unit or notation to another. This ensures that velocities can be easily compared and understood regardless of the chosen measurement system, such as feet per minute or meters per second. In the given exercise, the rocket's escape velocity was first provided in scientific notation. Converting this to standard notation made it easier to visualize the speed magnitude. Such conversions are vital in many practical scenarios, like engineering, where precise speed measurements are key to designs or tests. To convert effectively:

• First interpret the initial value.
• Identify the new desired format or unit.
• Perform necessary calculations to achieve the format.

Understanding velocity conversion can simplify complex data and enhance precision in scientific and engineering tasks.