Certainly! Let's go through the solution step-by-step.
1. Understanding the problem:
We have a right rectangular prism (a three-dimensional figure with length [tex]\(l\)[/tex], width [tex]\(w\)[/tex], and height [tex]\(h\)[/tex]). The volume [tex]\(V\)[/tex] of this prism is given as [tex]\(60 \, \text{cm}^3\)[/tex].
2. Volume of a rectangular prism:
The volume of a rectangular prism is calculated using the formula:
[tex]\[
V = l \times w \times h
\][/tex]
3. Given volume:
[tex]\[
60 \, \text{cm}^3 = l \times w \times h
\][/tex]
4. Finding suitable dimensions [tex]\(l\)[/tex], [tex]\(w\)[/tex], and [tex]\(h\)[/tex]:
There are multiple sets of dimensions [tex]\( (l, w, h) \)[/tex] that can satisfy the equation [tex]\(l \times w \times h = 60 \, \text{cm}^3\)[/tex]. Here, we assume the values for [tex]\(l\)[/tex], [tex]\(w\)[/tex], and [tex]\(h\)[/tex] that are positive numbers. Let's choose straightforward values to make the calculation simple. We can take:
Let [tex]\(l = 5 \, \text{cm}\)[/tex], [tex]\(w = 3 \, \text{cm}\)[/tex], and [tex]\(h = 4 \, \text{cm}\)[/tex].
5. Checking our chosen dimensions:
[tex]\[
5 \, \text{cm} \times 3 \, \text{cm} \times 4 \, \text{cm} = 60 \, \text{cm}^3
\][/tex]
Our choice is correct because it satisfies the given volume condition.
6. Determine the area of the base:
The base of the right rectangular prism can be any of the three possible pairs of sides. Commonly, we consider the area formed by length and width as the base.
[tex]\[
\text{Area of the base} = l \times w = 5 \, \text{cm} \times 3 \, \text{cm} = 15 \, \text{cm}^2
\][/tex]
7. Final Results:
- Length [tex]\( l = 5 \, \text{cm} \)[/tex]
- Width [tex]\( w = 3 \, \text{cm} \)[/tex]
- Height [tex]\( h = 4 \, \text{cm} \)[/tex]
- Area of the base [tex]\( = 15 \, \text{cm}^2 \)[/tex]
Thus, the length, width, and height of the prism are [tex]\(5 \, \text{cm}\)[/tex], [tex]\(3 \, \text{cm}\)[/tex], and [tex]\(4 \, \text{cm}\)[/tex] respectively. The area of the base is [tex]\(15 \, \text{cm}^2\)[/tex].
