Question

The first Apple computer had \(5.0\) Mbytes of storage space on its hard drive, and the cost of this computer was \(\$ 9995\). An Apple iPad Air 2 has 128 Gbytes of storage for a cost of S699. Calculate the cost per byte for each of these two Apple products. Which is a better buy?

Short answer

The iPad Air 2 is a better buy at 5.46\( \times 10^{-6} \) dollars/byte compared to the Apple computer at 0.001999 dollars/byte.

Step-by-step solution

- Convert Storage Units for Apple Computer

First, convert the storage space of the original Apple computer from Mbytes to bytes. We know that 1 Mbyte = 1,000,000 bytes. So, the storage space of the Apple computer in bytes is: \[5.0 \text{ Mbytes} \times 1,000,000 \text{ bytes/Mbyte} = 5,000,000 \text{ bytes}\]

- Calculate Cost Per Byte for Apple Computer

Next, calculate the cost per byte for the Apple computer by dividing its cost by its storage space: \[\frac{\$9995}{5,000,000 \text{ bytes}} = 0.001999 \text{ dollars/byte}\]

- Convert Storage Units for Apple iPad Air 2

Now, convert the storage space of the iPad Air 2 from Gbytes to bytes. We know that 1 Gbyte = 1,000,000,000 bytes. So, the storage space of the iPad in bytes is: \[128 \text{ Gbytes} \times 1,000,000,000 \text{ bytes/Gbyte} = 128,000,000,000 \text{ bytes}\]

- Calculate Cost Per Byte for Apple iPad Air 2

Next, calculate the cost per byte for the iPad Air 2 by dividing its cost by its storage space: \[\frac{\$699}{128,000,000,000 \text{ bytes}} = 5.46 \times 10^{-6} \text{ dollars/byte}\]

- Compare Costs Per Byte

Compare the cost per byte for both products. The Apple computer has a cost per byte of 0.001999 dollars/byte, whereas the iPad Air 2 has a cost per byte of 5.46\( \times 10^{-6} \) dollars/byte. Thus, the iPad Air 2 is a better buy.

Key concepts

Unit Conversion

Understanding unit conversion is essential for calculating the cost per byte for both devices. To convert units, you need to know the conversion factors. In this case, 1 megabyte (Mbyte) equals 1,000,000 bytes, and 1 gigabyte (Gbyte) equals 1,000,000,000 bytes.

• For the original Apple computer: Multiply 5 Mbytes by 1,000,000 to get 5,000,000 bytes.
• For the iPad Air 2: Multiply 128 Gbytes by 1,000,000,000 to get 128,000,000,000 bytes.

Using these conversions, we can calculate the storage space in bytes, which is crucial for further calculations.

Cost Efficiency Comparison

To determine the better buy, you compare the cost per byte of storage for each device. Calculate cost per byte by dividing the device cost by its storage space in bytes.

• For the original Apple computer: \(\frac{\text{\textdollar}9995}{5,000,000 \text{ bytes}}\) equals approximately 0.001999 dollars per byte.
• For the iPad Air 2: \(\frac{\text{\textdollar}699}{128,000,000,000 \text{ bytes}}\) equals approximately 5.46 \(\times 10^{-6}\) dollars per byte.

By comparing these values, we see that the iPad Air 2 has a significantly lower cost per byte, making it more cost-efficient.

Storage Space Calculation

Accurate storage space calculation is vital for understanding data capacity of electronic devices. For this exercise:

• The storage space of the Apple computer is calculated as 5 Mbytes \( \times 1,000,000 \) to get 5,000,000 bytes.
• The storage space of the iPad Air 2 is 128 Gbytes \( \times 1,000,000,000 \) to get 128,000,000,000 bytes.

By converting Mbytes and Gbytes to bytes, we can accurately compare the storage capacities of both devices.

Problem Solving in Chemistry Education

Problem-solving skills gained in chemistry education are essential for tackling diverse challenges. These skills develop through exercises like this one, involving unit conversion and cost calculations. Chemistry students learn to:

• Break complex problems into manageable steps.
• Use systematic approaches to find solutions.
• Apply theoretical knowledge to practical scenarios.

These skills are transferable across various scientific and technical fields, enhancing overall problem-solving capabilities.

Practical Applications of Chemistry in Technology

The principles of chemistry are often used in technology, helping us understand and utilize materials better. For example, unit conversion and cost efficiency calculations:

• Help in determining the economic value of technological devices.
• Allow manufacturers to optimize resource usage.
• Aid consumers in making informed purchasing decisions.

These applications demonstrate the relevance of chemistry beyond the lab, influencing everyday technologies like computers and tablets.