Question

The cumulonimbus cloud pictured in Figure \(17.21 \mathrm{H}\) is roughly 12 kilometers tall, 8 kilometers wide, and 8 kilometers long. Assume that the droplets in each cubic meter of the cloud total 0.5 cubic centimeter. How much liquid does the cloud contain? How many gallons is this? (Note: \(3785 \mathrm{~cm}^{3}=1\) gallon.)

Short answer

The cloud contains approximately 1.014 × 10^8 gallons of water.

Step-by-step solution

Calculate the Volume of the Cloud

First, find the volume of the cumulonimbus cloud given its dimensions. The volume of a rectangular prism is determined by multiplying its height, width, and length. The cloud is 12 kilometers tall, 8 kilometers wide, and 8 kilometers long. Use the formula: \[ V = \text{height} \times \text{width} \times \text{length} \]Convert these dimensions into meters (since 1 km = 1000 m):\[ \text{Height} = 12 \text{ km} = 12000 \text{ m} \]\[ \text{Width} = 8 \text{ km} = 8000 \text{ m} \]\[ \text{Length} = 8 \text{ km} = 8000 \text{ m} \]Then, calculate the volume:\[ V = 12000 \times 8000 \times 8000 = 7.68 \times 10^{11} \text{ m}^3 \]

Calculate the Total Liquid Volume in Cubic Meters

Given that each cubic meter contains 0.5 cubic centimeters of droplets, find the total volume of liquid in the entire cloud. Since there are \(7.68 \times 10^{11}\) cubic meters in the cloud, multiply this by the volume of droplets per cubic meter:\[ \text{Total liquid volume} = 7.68 \times 10^{11} \times 0.5 \text{ cm}^3 \]\[ \text{Total liquid volume} = 3.84 \times 10^{11} \text{ cm}^3 \]

Convert Cubic Centimeters to Gallons

Convert the calculated cubic centimeters of liquid into gallons using the conversion factor \(3785 \, \text{cm}^3 = 1\) gallon.Use the formula:\[ \text{Gallons} = \frac{\text{Total liquid volume in cm}^3}{3785} \]\[ \text{Gallons} = \frac{3.84 \times 10^{11}}{3785} \approx 1.014 \times 10^{8} \text{ gallons} \]

Conclude with the Total Liquid Volume

Summarize the amount of liquid the cloud contains in gallons based on the previous calculations. The cumulonimbus cloud contains approximately \(1.014 \times 10^{8}\) gallons of water.

Key concepts

Volume of Cloud

Understanding how to calculate the volume of a cloud is an exciting journey into the world of meteorological calculations. In this case, imagine a large cumulonimbus cloud with the shape of a rectangular prism. To find the cloud's volume, you use the formula for the volume of a prism:\[ V = \text{height} \times \text{width} \times \text{length} \]
First, you have to convert all dimensions from kilometers to meters, since our final volume will be in cubic meters. Keep in mind:

• Height: 12 km becomes 12,000 meters
• Width: 8 km becomes 8,000 meters
• Length: 8 km becomes 8,000 meters

By using these conversions, the calculation turns into multiplying \(12,000 \times 8,000 \times 8,000\), giving a total volume of \(7.68 \times 10^{11}\) cubic meters. That is a massive cloud indeed!

Liquid Water Content

The next step is discovering how much actual liquid water is contained within all that cloud volume. Clouds are mainly composed of tiny droplets of water suspended in the air. Our cloud has a liquid water content, meaning it contains droplets that sum up to 0.5 cubic centimeters per cubic meter.
To find the total amount of liquid, multiply the number of cubic meters in the cloud by the liquid water content per cubic meter. That means:\[ \text{Total liquid volume} = 7.68 \times 10^{11} \times 0.5 \text{ cm}^3 \]
This calculation results in \(3.84 \times 10^{11}\) cubic centimeters of water droplets dispersed throughout the cloud's volume. It helps visualize how much rain these clouds may produce.

Metric to Imperial Conversion

In science and everyday applications, converting between metric and imperial units is essential. Here, you need to convert from cubic centimeters (metric) to gallons (imperial). Knowing the exact relationship between the two units is crucial for accurate conversion. For this specific exercise:

• 1 cubic meter equals 1,000,000 cubic centimeters
• 1 gallon equals 3,785 cubic centimeters

Utilizing the conversion factor \(3,785 \text{ cm}^3 = 1\) gallon, transforms our calculation from cubic centimeters, which are part of the metric system, to gallons, an imperial measurement. These conversions are often vital in comparing quantities worldwide since different regions use different systems.

Cubic Meter to Gallons Conversion

To find out how many gallons are present in the cloud, you need to convert the total volume of water droplets from cubic centimeters to gallons. You have the total liquid volume calculated in cubic centimeters as \(3.84 \times 10^{11} \text{ cm}^3\).
Using the conversion formula: \[\text{Gallons} = \frac{3.84 \times 10^{11}}{3,785}\]
This division gives approximately \(1.014 \times 10^{8}\) gallons. Understanding this conversion helps you relate physically to the amount of water present in the cloud. Imagine countless buckets of water making up a tremendous reservoir in the sky, showcasing nature's capacity for moisture storage and distribution in easily relatable terms.