https://file.aiquickdraw.com/mathbot/file/1728716391700zj0j6vbl.jpg

To solve the problem, let's denote the dimensions of the rectangular box as \( x \), \( y \), and \( z \). The areas of the three faces of the rectangular box are given as follows:

\[
xy = 30 \, \text{cm}^2, \quad yz = 18 \, \text{cm}^2, \quad xz = 15 \, \text{cm}^2
\]

We are asked to find the volume of the box, which is \( V = x \times y \times z \).

### Step 1: Multiply all the area equations
To eliminate the variables, multiply the three equations together:

\[
(xy)(yz)(xz) = 30 \times 18 \times 15
\]

Simplifying the left-hand side:

\[
(xy)(yz)(xz) = (x \times y \times z)^2 = V^2
\]

Now, simplify the right-hand side:

\[
30 \times 18 \times 15 = 8100
\]

So, we have:

\[
V^2 = 8100
\]

### Step 2: Solve for the volume
To find \( V \), take the square root of both sides:

\[
V = \sqrt{8100} = 90 \, \text{cm}^3
\]

Thus, the volume of the box is \( \boxed{90} \, \text{cm}^3 \).

Would you like further details on any step, or have any questions?

---

Here are five follow-up questions to explore further:
1. What are the individual dimensions \( x \), \( y \), and \( z \) of the box?
2. How would the solution change if one of the given areas was different?
3. Can you find the surface area of the box based on the given information?
4. What is the geometric significance of multiplying the areas of the faces?
5. How would you approach the problem if only two areas were given?

**Tip**: When solving for dimensions, multiplying related variables and simplifying equations can help eliminate unnecessary variables.

Three of the faces of a rectangular box have areas of 30 cm², 18 cm², and 15 cm². What is the volume (in cm³) of the box?

Learn how to calculate the volume of a rectangular box when the areas of its three faces are 30 cm², 18 cm², and 15 cm². This geometry problem covers key concepts such as volume, rectangular box dimensions, and area multiplication.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation