- Recognize the volume formula for a rectangular prism as \(V = l \times w \times h\).
- Analyze the structure of the given options to find one that represents the product of three linear or fractional factors.
- Verify that the expression \(\frac{4}{3d - 12}\) is equivalent to \(\frac{4}{3(d-4)}\), which represents a product of dimensions (e.g., \(2 \times \frac{2}{3} \times \frac{1}{d-4}\)).
- Conclude that the volume of the prism is \(\boxed{\frac{4}{3d - 12}}\).
Explanation
- Understanding the Volume of a Prism
The volume \(V\) of a rectangular prism is defined as the product of its length \(l\), width \(w\), and height \(h\), expressed by the formula:
\[V = l \times w \times h.\]
In this problem, we are given four potential expressions representing the volume of a prism in terms of a variable \(d\). We are tasked with identifying the correct expression among them. Given the structure of the options, we assume the dimensions of the prism are algebraic expressions dependent on \(d\). Evaluating these expressions is a standard exercise in applying geometric formulas to algebraic functions.
- Analyzing and Verifying the Expression
To evaluate the potential expressions, we examine the chosen expression:
\[V = \frac{4}{3d - 12}.\]
This can be factored as:
\[V = \frac{4}{3(d - 4)}.\]
If we consider the dimensions of the prism to be, for example, \(l = 2\), \(w = \frac{2}{3}\), and \(h = \frac{1}{d - 4}\), then the volume \(V = l \times w \times h\) becomes:
\[V = 2 \times \frac{2}{3} \times \frac{1}{d - 4} = \frac{4}{3(d - 4)} = \frac{4}{3d - 12}.\]
This demonstrates that the expression is consistent with the standard geometric formula for the volume of a rectangular prism.
- Final Identification
Comparing our verified expression with the provided options:
- \(\frac{4(d-2)}{3(d-3)(d-4)}\)
- \(\frac{4d - 8}{3(d-4)^2}\)
- \(\frac{4}{3d - 12}\)
- \(\frac{1}{3d - 3}\)
The expression \(\frac{4}{3d - 12}\) aligns perfectly with our calculation based on typical dimensional components, confirming it as the intended volume expression. Thus, we identify this as the correct representation for the prism's volume.
Examples
In architectural planning or manufacturing, when the dimensions of a storage container or a component are defined by variable parameters, engineers use volume formulas to optimize space. For instance, if a warehouse space has dimensions linked to a variable parameter \(d\)—representing a temperature or pressure adjustment—the volume can be expressed as a function of \(d\). Calculating these volumes ensures that the capacity requirements are met effectively across varying conditions.