Problem 81 - Algebra A Combined Function

Question

As of this writing, the world's largest optical telescope is the Gran Telescopio Canaris, located in La Palma, Canary Islands, Spain. The elevation of this telescope is 2400 meters above sea level. Write 2400 in scientific notation.

Step-by-Step Solution

Answer

2400 in scientific notation is \(2.4 \times 10^3\).

Step 1: Identify the Number

The number we need to convert to scientific notation is 2400.

Step 2: Understand Scientific Notation

Scientific notation expresses a number as the product of a number between 1 and 10 and a power of 10. The general form is \(a \times 10^n\).

Step 3: Find the Coefficient 'a'

To convert 2400 to scientific notation, we first find the coefficient \(a\) by moving the decimal point so that the number is between 1 and 10. For 2400, this becomes 2.4.

Step 4: Determine the Power of Ten 'n'

Determine how many places we need to move the decimal point to position it just after the first digit. In 2400, the decimal is moved 3 places to the left: 2400 -> 2.4. Hence, \(n = 3\).

Step 5: Combine into Scientific Notation

Combine \(a = 2.4\) with \(10^n\), where \(n = 3\). Thus, 2400 in scientific notation is \(2.4 \times 10^3\).

Key Concepts

Optical Telescope

An optical telescope is a fascinating instrument designed to observe distant objects. It collects and magnifies the light from stars, planets, and other celestial bodies. These telescopes use lenses or mirrors to focus and magnify light. They are essential tools in astronomy.

Key components include:

• Objective Lens or Mirror: Captures light and forms an image.
• Eyepiece: Magnifies the image for detailed observation.
• Mount: Provides stability and allows for accurate aiming.

Optical telescopes come in various sizes, from small personal ones to large, ground-based observatories like the Gran Telescopio Canaris. The larger the telescope, the more light it can gather, enhancing its ability to study faint objects.

Telescope positioning at high elevations, such as the Gran Telescopio Canaris at 2400 meters, helps reduce atmospheric interference. This location offers clearer views and allows astronomers to study the universe in incredible detail.

Power of Ten

The 'power of ten' is a concept that helps simplify working with very large or very small numbers. It is essentially the number of times you multiply 10 by itself. This method is part of what's called 'exponentiation'.

For example:

• The expression \(10^3\) means you multiply 10 by itself three times: \(10 imes 10 imes 10 = 1000\).
• For 10 to the power of one, \(10^1\) is just 10.

Using the power of ten makes it easier to express and calculate with large numbers. In our case of converting 2400, we recognize it as 2.4 with the power of ten being 3 (since moving the decimal three places makes 2400). Understanding how to use powers of ten efficiently is crucial in mathematical calculations, especially in scientific fields.

Conversion to Scientific Notation

Converting a number to scientific notation is a helpful way to express large or small numbers succinctly. This format shows a number as the product of two factors:

• A number between 1 and 10.
• A power of ten.

To convert 2400 to scientific notation, follow these steps:

First, identify where to place the decimal to form a number between 1 and 10. For 2400, moving the decimal point two places left gives 2.4. This number, 2.4, is known as the coefficient.

The next step is to count how many places you moved the decimal point, which is 3 for 2400. This count becomes the power of ten: \(10^3\).

Finally, combine the coefficient and the power of ten to get the scientific notation: 2400 is written as \(2.4 \times 10^3\). This method makes it easier to read, understand, and compute with very large or small numbers.