Problem 106 - General Chemistry

Question

A spherical tank has a radius of 175.0 in. Calculate the volume of the tank in cubic inches; then convert this to Imperial gallons. The volume of a sphere is \((4 / 3) \pi r^{3},\) where \(r\) is the radius. One Imperial gallon equals 277.4 in \(^{3}\).

Step-by-Step Solution

Answer

81025.6 Imperial gallons

Step 1: Calculate the Volume of the Sphere

The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. Substitute \( r = 175.0 \) inches into the formula. Thus, \( V = \frac{4}{3} \pi (175.0)^3 \). Calculating this gives \( V = \frac{4}{3} \pi \times 5359375 \). Approximating \( \pi \) as 3.1416, the volume is \( V \approx 22478875.0 \, \text{cubic inches}\).

Step 2: Convert Cubic Inches to Imperial Gallons

Since 1 Imperial gallon is equal to 277.4 cubic inches, divide the volume in cubic inches by 277.4 to convert it to Imperial gallons. Thus, \( \frac{22478875.0}{277.4} \approx 81025.6 \). So, the volume of the tank is approximately \( 81025.6 \, \text{Imperial gallons}\).

Key Concepts

Understanding the Spherical Tank

A spherical tank is a storage container shaped like a sphere. This shape is known for its aesthetic appeal and efficient material use. However, calculating its volume can initially seem challenging.

The formula for finding the volume of a sphere is \[V = \frac{4}{3} \pi r^3\]Here, \( V \) represents the volume, \( \pi \) is approximately 3.1416, and \( r \) is the radius. The radius is the distance from the center of the sphere to its surface.

To find the volume of a spherical tank, plug the radius into this formula. Make sure your measurements are consistent, focusing on using the same unit throughout, like in this case, inches. This approach allows you to accurately determine the capacity of a spherical tank in terms of volume.

What Are Imperial Gallons?

Imperial gallons are a unit of liquid volume commonly used in the UK and other British Commonwealth countries. It's crucial when dealing with international or diverse measurements to know how to convert to Imperial gallons from cubic inches.

One Imperial gallon is defined as 277.4 cubic inches. This distinction is important, as it differs from the US gallon measurement of volume. Being aware of this difference helps in converting and measuring across various systems, essential for applications in engineering and science.

• Used in the UK, Canada, and other Commonwealth nations
• One Imperial gallon = 277.4 cubic inches
• Different from the US gallon

Relating these values back to your calculation ensures precision when converting volume measures. This conversion plays a vital role when determining how much liquid a spherical tank can hold when measurements start in cubic inches.

Converting Cubic Inches to Imperial Gallons

Cubic inches are another unit of volume, and sometimes conversions are necessary when dealing with large volumes like that of a spherical tank.

To convert from cubic inches to Imperial gallons, you divide the total cubic inches by the volume equivalent of one Imperial gallon, which is 277.4 cubic inches.

For example, if a tank's volume is calculated at 22,478,875 cubic inches, dividing this by 277.4 will provide the volume in Imperial gallons.

Calculation would go as follows:

• Start with volume in cubic inches: 22,478,875 cubic inches
• Convert using 277.4 cubic inches per Imperial gallon
• Result: \( \frac{22,478,875}{277.4} \approx 81,025.6 \) Imperial gallons

Understanding this conversion process helps bridge the gap between different measurement systems, providing a clear, universal metric for fluid capacities.