Problem 79 - Algebra A Combined Function

Question

The formula \(V=l w h\) is used to find the volume of a box. If the length of a box is doubled, the width is doubled, and the height is doubled, how does this affect the volume? Explain your answer.

Step-by-Step Solution

Answer

The new volume is 8 times the original volume.

Step 1: Understanding the Original Volume Formula

The original formula to calculate the volume of a box is \( V = lwh \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height. This formula helps us understand how the dimensions of a box contribute to its volume.

Step 2: Applying the Changes to Dimensions

Each dimension of the box is doubled: the length becomes \( 2l \), the width becomes \( 2w \), and the height becomes \( 2h \). So, we look at how these changes affect the formula for the volume.

Step 3: Calculating the New Volume

Replace \( l \), \( w \), and \( h \) in the volume formula with the new dimensions: \( V' = (2l)(2w)(2h) \). Calculate this expression to find the new volume: \( V' = 8lwh \).

Step 4: Comparing the Original and New Volumes

The original volume was \( V = lwh \). The new volume is \( V' = 8lwh \), which is 8 times larger than the original volume. This demonstrates that doubling each dimension of a box results in the volume being increased by a factor of eight.

Key Concepts

geometry

When discussing the concept of geometry, we refer to the branch of mathematics that deals with shapes, sizes, and the properties of space. Geometry plays a fundamental role when it comes to understanding and calculating the volume of different three-dimensional shapes, such as boxes.

A box is a practical example of a rectangular prism, a common geometric shape. This is crucial when we want to calculate its volume or understand how changing its dimensions affects it. The box's geometric properties, like length, width, and height, define how much space it occupies. These dimensions are essential inputs in the volume formula, which gives us a numerical value representing the box's space or capacity.

volume formula

To find the volume of a box, the formula used is: \( V = lwh \), where:

• \( l \) stands for the length of the box.
• \( w \) represents the width.
• \( h \) denotes the height.

This formula is a straightforward multiplication of the three dimensions itself. In essence, it tells us how many unit cubes can fit inside that box.

For instance, if the length, width, and height of a box are all measured in centimeters, the volume will be in cubic centimeters. Understanding this formula is vital because the volume is a critical measurement in many real-world applications, like packing, shipping, or building projects.

effect of scaling dimensions

The effect of scaling dimensions in geometry can be profound, especially when considering volume. In our exercise, if the length, width, and height of a box are each doubled, this drastically impacts the box's volume.

When scaling each dimension by a factor of 2, we use the new formula: \( V' = (2l)(2w)(2h) \).
Simplifying, we find \( V' = 8lwh \).

This calculation shows that the new volume is eight times the original volume. By scaling each dimension, the volume doesn't just double or triple; it's multiplied by the cube of the scaling factor (in this case, 2³ = 8). Understanding this power of scaling can help make informed decisions in design and construction, where size and capacity are crucial considerations. Reconfiguring dimensions leads to exponential changes in volume, a vital concept for engineering, architecture, and other applied sciences.