Problem 37 - Prealgebra

Question

As you know, the volume \(V\) enclosed by a rectangular solid with length \(I,\) width \(w,\) and height \(h\) is \(V=I \cdot w \cdot h .\) Find \(V\) if: \(I=6\) yards, \(w=\frac{1}{2}\) yard, and \(h=\frac{1}{3}\) yard

Step-by-Step Solution

Answer

The volume is 1 cubic yard.

Step 1: Write Down the Volume Formula

The formula for the volume of a rectangular solid is given by \( V = I \cdot w \cdot h \), where \( I \) is the length, \( w \) is the width, and \( h \) is the height.

Step 2: Substitute the Given Values

We substitute the given values into the formula: \( I = 6 \) yards, \( w = \frac{1}{2} \) yard, and \( h = \frac{1}{3} \) yard. This gives us the following equation for the volume: \( V = 6 \cdot \frac{1}{2} \cdot \frac{1}{3} \).

Step 3: Multiply the Length and Width

First, multiply the length and width: \( 6 \cdot \frac{1}{2} = \frac{6}{2} = 3 \).

Step 4: Multiply the Result by the Height

Now, multiply the result from Step 3 by the height: \( 3 \cdot \frac{1}{3} = 1 \).

Step 5: Conclusion

The volume of the rectangular solid is \( 1 \) cubic yard.

Key Concepts

volume calculation

Calculating volume, especially for rectangles, is an important skill in geometry. To find the volume of a rectangular solid, you multiply three dimensions: the length, the width, and the height. This formula expresses how much space the object occupies.In mathematical terms, we use the formula:\[ V = l \cdot w \cdot h \]Where:

• \( l \) is the length
• \( w \) is the width
• \( h \) is the height

To ensure all measurements are consistent, they should be in the same unit before calculating volume. For example, if one measurement is in meters and another in centimeters, conversions are necessary. In our example:\( V = 6 \cdot \frac{1}{2} \cdot \frac{1}{3} \), all lengths are in yards, so the final volume is 1 cubic yard.

rectangular prism

A rectangular prism is a type of three-dimensional figure with opposite faces that are equal rectangles. They're incredibly common in both real-life scenarios and geometry problems.Key characteristics include:

• Six faces that are rectangles.
• Edges that meet at right angles.
• Identical opposite faces.

Rectangular prisms can vary greatly in shape and size but the method for calculating their volume remains unchanged. If the length is very small, such as in an extremely flat box, the prism becomes less of a tall block and more of a flat package, but the principle of calculating the volume is the same. The example given with lengths 6, \(\frac{1}{2}\), and \(\frac{1}{3}\) illustrates how volume calculations still apply even if one dimension is significantly smaller.

geometric formulas

Understanding and applying geometric formulas correctly is crucial for solving problems in mathematics. These formulas allow for precise calculations when working with geometric shapes, such as rectangles, circles, or other polygons.Volume for 3D shapes like the rectangular prism is just one category where geometric formulas apply. Here are other examples:

• The area of a rectangle: \( A = l \cdot w \)
• The circumference of a circle: \( C = 2\pi r \)
• The surface area of a sphere: \( A = 4\pi r^2 \)

By familiarity with these formulas, students can transition fluidly from working on one type of problem to another, transferring skills across different mathematical challenges.