Math Problem Statement

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?

Solution

To solve the problem, let's denote the dimensions of the rectangular box as xx, yy, and zz. The areas of the three faces of the rectangular box are given as follows:

xy=30cm2,yz=18cm2,xz=15cm2xy = 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×y×zV = x \times y \times z.

Step 1: Multiply all the area equations

To eliminate the variables, multiply the three equations together:

(xy)(yz)(xz)=30×18×15(xy)(yz)(xz) = 30 \times 18 \times 15

Simplifying the left-hand side:

(xy)(yz)(xz)=(x×y×z)2=V2(xy)(yz)(xz) = (x \times y \times z)^2 = V^2

Now, simplify the right-hand side:

30×18×15=810030 \times 18 \times 15 = 8100

So, we have:

V2=8100V^2 = 8100

Step 2: Solve for the volume

To find VV, take the square root of both sides:

V=8100=90cm3V = \sqrt{8100} = 90 \, \text{cm}^3

Thus, the volume of the box is 90cm3\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 xx, yy, and zz 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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Volume
Rectangular Box

Formulas

Volume of a rectangular box V = x * y * z
xy = 30 cm², yz = 18 cm², xz = 15 cm²
(x * y * z)² = (xy)(yz)(xz)

Theorems

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Suitable Grade Level

Grades 8-10