Problem 105 - General Chemistry

Question

An aquarium has a rectangular cross section that is 47.8 in. by 12.5 in.; it is 19.5 in. high. How many U.S. gallons does the aquarium contain? One U.S. gallon equals exactly \(231 \mathrm{in}^{3}\).

Step-by-Step Solution

Answer

The aquarium contains approximately 50.37 U.S. gallons.

Step 1: Calculate Volume of the Aquarium in Cubic Inches

The volume of the aquarium is found by multiplying its length, width, and height. Using the dimensions provided:\( \text{Volume} = 47.8 \, \mathrm{in} \times 12.5 \, \mathrm{in} \times 19.5 \, \mathrm{in} \).Calculating this gives:\( \text{Volume} = 11636.25 \, \mathrm{in}^3 \).

Step 2: Convert Cubic Inches to U.S. Gallons

To convert the volume from cubic inches to U.S. gallons, divide the volume by the number of cubic inches per gallon (231 in³/gallon):\( \text{Gallons} = \frac{11636.25 \, \mathrm{in}^3}{231 \, \mathrm{in}^3/\text{gallon}} \).Calculating this gives:\( \text{Gallons} \approx 50.37 \text{ gallons} \).

Key Concepts

Understanding a Rectangular Prism

A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. For practical purposes, it is recognizable as a box-like shape similar to everyday items such as a shoebox or, in this case, an aquarium. Understanding the dimensions of a rectangular prism is key to calculating various properties such as volume. In geometry, the dimensions refer to the length, width, and height of the prism. When you know these measurements, you can find the volume by multiplying them together.

• Length: How long the object is from one end to the other.
• Width: How wide the object is from side to side.
• Height: How tall the object is from bottom to top.

When dealing with a rectangular prism, like in this aquarium problem, you have to measure each of these dimensions to get the total volume, which plays a crucial role in further calculations.

Understanding Cubic Inches

A cubic inch is a unit of measure for volume in the imperial system, commonly used in the United States. One cubic inch refers to the volume of a cube with each side measuring exactly one inch. In problems involving volume, it's crucial to understand how to work with cubic inches, especially when converting units.To calculate volume in cubic inches, you multiply the object's dimensions (length, width, height) when they are also measured in inches. This results in a unit termed "cubic inch." It is written as \(\text{in}^3\), where the exponent 3 signifies a three-dimensional measurement.Using cubic inches is essential in various geometry problems, especially when you need to convert different units, as it serves as a basis for many volume conversions, including conversion to gallons in the US measurement system.

Converting to U.S. Gallons

In real-life applications, understanding how to convert cubic inches to U.S. gallons is highly beneficial. One U.S. gallon is equivalent to exactly 231 cubic inches, a figure that must be memorized or kept handy for conversions. When you have a volume in cubic inches, the conversion process is straightforward. You divide the total cubic inches by 231 to find the volume in U.S. gallons. For example, with an aquarium holding 11636.25 cubic inches, dividing by 231 gives approximately 50.37 gallons. This conversion is particularly useful in industries where liquid volumes are often represented in gallons, such as in aquariums, fuel tanks, and other fluid containers. Understanding these conversions helps in planning and efficiently using materials according to the U.S. system of measurement.

Geometry Applications in Daily Life

Geometry is everywhere around us, from the design of buildings to the sizing of a fish tank. By understanding geometric principles, like those of a rectangular prism, one can apply this knowledge to solve practical problems. Calculating the volume of an aquarium is an excellent example of applying geometry in real life. When purchasing or designing an aquarium, knowing its volume is essential to figure out how much water can be held and to plan for fish capacity, maintenance, and even the variety of aquatic life it can support.

• Helps determine the amount of water treatment needed.
• Assists in calculating the weight of the filled aquarium.
• Used to ensure structural support can withstand the full weight.

Geometry applications extend beyond just volume calculations and can aid in efficient architectural design, allowing for precise material use and aesthetic considerations, thus impacting both functional and visual aspects of construction.