Question

MAKING AN ARGUMENT The volume of a cube with side length \(x\) is given by \(V(x)=x^3\). Your friend claims that when you divide the volume in half, the volume decreases by a greater amount than when you divide each side length in half. Is your friend correct? Justify your answer.

Short answer

Yes, your friend is correct. The decrease in volume by halving the side lengths is greater than when halving the volume directly. This is evident from the fact that \(x^3/2 > x^3/8\)

Step-by-step solution

Calculate the volume when halving the sides

If you half the side length of the cube, the new side length will be \(x/2\). The new volume \(V'(x)\) will be \((x/2)^3 = x^3/8\)

Compare with half the original volume.

Now, let's look at what happens when we directly half the original volume. Half of the original volume is \(V(x)/2 = x^3/2\)

Compare the results

Upon comparing, we find that \(V(x)/2 > V'(x)\) or in other words \(x^3/2 > x^3/8\). So, halving the side length results in a volume less than half of the original volume

Conclude the argument

From steps 2 and 3, it's clear that your friend is correct. Halving the volume directly leads to a volume of \(x^3/2\) which is larger than the volume obtained by halving the side length (\(x^3/8\)). So, dividing the volume in half doesn't decrease the volume 'more' than halving each side length

Key concepts

Algebraic Reasoning

When we delve into algebraic reasoning, we're essentially engaging with the process of forming and solving algebraic expressions or equations based on a given problem. Consider the task of halving the volume of a cube, for example. Initially, it's necessary to understand the fundamental algebraic expression for the volume of a cube, which is given by the formula V(x) = x^3.

This equation serves as the foundation for any volume-related manipulations we aim to explore.Using algebraic reasoning, we take the side length, represented by x, and reformulate it when halved to x/2. Subsequently, to ascertain the new volume, we cube this new side length, arriving at the expression (x/2)^3. Such an approach exemplifies algebraic reasoning: taking an established algebraic structure and altering it methodically to resolve the problem at hand.

Geometric Properties

When we talk about geometric properties, we're referring to the attributes and characteristics that define various geometric shapes. In the case of a cube, a key property is that all sides are equal in length. This uniformity of side lengths simplifies the process of calculating volume, which is the amount of three-dimensional space enclosed by the cube, and is directly proportional to the cube of the side lengths.Understanding these properties allows us to predict that when we modify the side length of a cube, the volume will change quite drastically since the volume is cubed, not simply tripled or quadrupled. A cube’s volume will therefore always be sensitive to even small changes in side length due to its geometric properties.

Volume Calculation

Volume calculation for a cube is straightforward due to its regular shape. The volume is found by raising the side length to the third power, as mentioned earlier with the formula V(x) = x^3. It's essential, though, to thoroughly understand what happens when changes occur to the side length. Halving the side length as per the exercise doesn't halve the volume but reduces it by a ratio of 1:8. This is because when halving each individual dimension of a cube, we are essentially dividing the volume by 2 three times (once for each dimension), resulting in V(x)/8 rather than V(x)/2.This indicates that the relationship between a cube's side length and its volume is not linear but cubic, which is a pivotal concept in volume calculations involving cubes.

Mathematical Argumentation

Mathematical argumentation involves presenting a logical series of statements to support a mathematical claim. In our scenario, proving whether halving the volume or the side length decreases the cube's volume by a greater amount requires a structured argument.To create this argument, first, we calculate the volume of the cube with a halved side length, V'(x). Next, we directly halve the cube's volume, V(x)/2. By comparing these values, V(x)/2 and V'(x), we can observe that V(x)/2 is indeed greater than V'(x). This comparison is the crux of our mathematical argument, which supports the friend's claim with clear, logical evidence derived from our previous algebraic and geometric analysis.