Question

Suppose a machine has 200 GB of storage space available on a hard disk and receives data over a broadband connection at the rate of \(15 \mathrm{Mbps}\). At this rate, how long will it take to fill the available storage space?

Short answer

It will take approximately 29.63 hours to fill the storage space.

Step-by-step solution

Convert storage space to bits

The storage space available is 200 GB. To do calculations with the data transfer rate, convert this space into bits. First, convert gigabytes to bytes: \[ 200 \text{ GB} = 200 \times 10^9 \text{ bytes} \]Then convert bytes to bits (since there are 8 bits in 1 byte):\[ 200 \times 10^9 \text{ bytes} = 200 \times 10^9 \times 8 \text{ bits} = 1.6 \times 10^{12} \text{ bits} \]

Convert data rate to bits per second

The data rate is given as \(15 \text{ Mbps}\). Convert this to bits per second:\[ 15 \text{ Mbps} = 15 \times 10^6 \text{ bps} \]

Calculate time to fill storage

We need to calculate the time it takes in seconds to fill the storage space at the given rate of 15 Mbps. To find the time, divide the total number of bits by the data transfer rate:\[ \text{Time (in seconds)} = \frac{1.6 \times 10^{12} \text{ bits}}{15 \times 10^6 \text{ bps}} = \frac{1.6}{15} \times 10^6 \]This simplifies to:\[ \frac{1.6}{15} \times 10^6 \approx 106666.67 \text{ seconds} \]

Convert time to hours

Finally, convert the time from seconds to hours. Since there are 3600 seconds in an hour:\[ \text{Time (in hours)} = \frac{106666.67}{3600} \approx 29.63 \text{ hours} \]

Key concepts

Storage Capacity

When discussing data storage, it's crucial to understand the concept of 'storage capacity.' This capacity indicates how much data can be held by a storage device, like a hard disk, USB drive, or memory card. Let's take a look at some basic conversion steps to grasp how storage capacity is interconnected with various units:

• Bytes and Bits: Storage is typically measured in bytes. One byte equals 8 bits. Thus, when dealing with higher units such as gigabytes or terabytes, it's common to convert these into bits for detailed calculations.
• Gigabytes (GB) to Bytes: 1 GB is equal to 10⁹ bytes. To convert from gigabytes to bytes, multiply the gigabytes by 10⁹.
• Bytes to Bits: Convert bytes to bits by multiplying the byte count by 8 since 1 byte = 8 bits.

By converting the storage capacity to bits, it allows us to directly relate it to the data transfer rate, making further calculations straightforward, as we examine how much time is needed to fill this storage.

Conversion of Units

In order to solve problems involving different data units, it's essential to be good at converting them accurately. The process involves switching between various units such as megabits, bits, and bytes, adapting them for coherent and correct calculations. Here's how you can manage these conversions:

• Understand Units: Familiarize yourself with terms like bytes, kilobytes, megabytes, and their bit counterparts. Remember, each step up the scale involves multiplying by powers of 10.
• Data Transfer Rate: Measured in bits per second (bps), often represented in higher terms like Megabits per second, Mbps, where 1 Mbps equals 10⁶ bps.
• Simple Conversion: If the data rate is provided in Mbps, convert it into bps before proceeding with calculations, by multiplying the Mbps value by 10⁶.

Being adept at these unit conversions helps you compute times, speeds, and capacities, especially when analyzing how data flows or fills a storage device like a hard disk.

Seconds to Hours Conversion

After finding out the time in seconds, especially in scenarios like measuring data transfer duration or filling a storage unit, converting into hours can offer clearer understanding. Here's a simple insight into the conversion process:

• Basics of Time Units: There are 60 seconds in a minute and 60 minutes in an hour. Hence, you have a total of 3600 seconds in one hour.
• Convert Seconds to Hours: Divide the total seconds by 3600 to get the equivalent time in hours. This kind of calculation simplifies understanding the length of time involved in extended durations, like waiting for a file to download or a storage medium to fill.
• Practical Use: For example, if a calculation gives 106,666.67 seconds, you can easily find it's approximately 29.63 hours by dividing by 3600.

This conversion is a straightforward yet powerful tool for visualizing time spans in everyday units we're more familiar with, such as hours or minutes.