Solving the Puzzle of a Box's Dimensions and Volume

The problem of determining the dimensions and volume of a cuboid based on its surface area is a classic example in geometry. In this article, we will explore how to solve such problems step by step using a specific example. We will also provide a detailed explanation of the mathematical formulas and steps involved.

Problem Statement

Given the edges of a rectangular box are in the ratio 1:2:4 and its surface area is 88 cm2. What is the volume of the box?

Understanding the Problem

First, let's define the variables and equations needed to solve this problem. We know that the edges of the cuboid are in the ratio 1:2:4, and the surface area is given.

Let the dimensions of the cuboid be:

Length (L) x Breadth (B) 2x Height (H) 4x

The formula for the surface area of a cuboid is:

Surface Area (SA) 2(LB BH HL)

Given the surface area (SA) 88 cm2, we can set up the equation:

88 2(4x times; 2x 2x times; 4x 4x times; x)

Solving for x

Simplifying the equation:

88 2(8x2 8x2 4x2)

88 2(22)

88 42

x2 88 / 40

x2 2.2

x √2.2

Now that we have x, we can find the dimensions of the cuboid:

Length, L x √2.2 ≈ 1.483 cm Breadth, B 2x 2 times; √2.2 ≈ 2.966 cm Height, H 4x 4 times; √2.2 ≈ 5.932 cm

Volume Calculation

To calculate the volume (V) of the cuboid, we use the formula:

Volume (V) L times; B times; H

Substituting the values:

V ≈ 1.483 times; 2.966 times; 5.932 ≈ 27.242 cm3

Therefore, the volume of the cuboid is approximately 27.242 cm3.

General Solution Approach

The general approach to solving such problems involves:

Define the dimensions of the cuboid using the given ratio. Set up the equation for the surface area based on the dimensions. Solve the equation to find the value of the unknown variable. Use the value of the variable to find the actual dimensions. Calculate the volume using the dimensions.

Conclusion

Understanding and applying the formula for the surface area of a cuboid is crucial in solving geometric problems. By breaking down the problem into manageable steps and using algebraic manipulation, we can efficiently find the dimensions and volume of the cuboid. This problem-solving technique is not only useful in mathematics but also in various real-world applications involving spatial dimensions.