Problem 38 - 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: \(I=30\) yards, \(w=\frac{5}{2}\) yards, and \(h=\frac{5}{3}\) yards

Step-by-Step Solution

Answer

The volume \( V \) is 125 cubic yards.

Step 1: Identify Given Values

We are given the length \( I = 30 \) yards, the width \( w = \frac{5}{2} \) yards, and the height \( h = \frac{5}{3} \) yards.

Step 2: Write the Formula for Volume

The volume \( V \) of a rectangular solid is calculated using the formula \( V = I \cdot w \cdot h \).

Step 3: Substitute Values into the Formula

Substitute the given values into the formula: \( V = 30 \cdot \frac{5}{2} \cdot \frac{5}{3} \).

Step 4: Simplify the Multiplication

First, multiply \( 30 \cdot \frac{5}{2} = 30 \cdot 2.5 = 75 \). Next, multiply \( 75 \cdot \frac{5}{3} = 75 \cdot 1.6667 = 125 \).

Step 5: Conclusion

The volume of the rectangular solid is \( V = 125 \) cubic yards.

Key Concepts

Volume calculation

Understanding how to calculate the volume of different geometric shapes is a fundamental aspect of prealgebra. The volume of an object is the amount of space it occupies. For a rectangular solid, or a box-shaped figure, the volume can be found using a simple formula: \[ V = I \cdot w \cdot h \] where:

• \( V \) represents the volume of the solid.
• \( I \) is the length of the solid.
• \( w \) is the width.
• \( h \) is the height.

In this context, volume is measured in cubic units. For example, if the dimensions are in yards, like in the problem we worked on, the volume will be in cubic yards.
Calculating volume involves multiplying these three dimensions together, which can be thought of as stacking layers of the shape's base over its height, filling up space.

Rectangular solid

A rectangular solid, also known as a rectangular prism, is a three-dimensional shape with six rectangular faces. These faces are organized so that opposite faces are parallel and equal in size. This type of solid is extremely common in various applications, from simple storage boxes to complex architectural structures. The key properties of a rectangular solid include:

• Six faces - all rectangles.
• Twelve edges - the sides of these faces.
• Eight vertices - where the edges meet.

Regardless of its specific dimensions, calculating the volume of a rectangular solid always follows the same principle: multiply its length, width, and height. This consistent formula helps when dealing with different size boxes and comparing their capacities.
Because it's one of the basic shapes, understanding how to work with rectangular solids is crucial for more advanced geometry scenarios.

Step-by-step math solution

Approaching math problems using a step-by-step process makes the problem-solving journey easier and more organized. First, identify given values. Knowing exactly what you have in terms of numbers and variables sets a solid foundation for solving problems.The next step involves writing down the relevant formula, which in this case is the volume formula for a rectangular solid: \[ V = I \cdot w \cdot h \].Substitute the known values into this equation. For the problem given, it's substituting the length, width, and height with their respective values: 30, \( \frac{5}{2} \), and \( \frac{5}{3} \).
Once substituted, simplify the equation by performing the multiplication:

• First, compute \( 30 \times \frac{5}{2} \)
• Then multiply the result by \( \frac{5}{3} \)

The simplification step often involves intermediate calculations where fractions are involved, aiding in gradually arriving at the final result.
Finally, interpret and conclude the calculations with the resulting volume measurement. Following this structured approach not only simplifies the task but also builds confidence in tackling similar mathematical challenges.