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Chapter 49: Problem 853

If a cube has edge length a, what happens to (a) the cube's surface area as a is increased by a factor \(x ?\) b) the cube's volume as a is increased by a factor \(\mathrm{x}\) ?

Short Answer

Expert verified
a) When the edge length of a cube is increased by a factor of x, the surface area is increased by a factor of \(x^2\). b) When the edge length of a cube is increased by a factor of x, the volume is increased by a factor of \(x^3\).

Step by step solution

01

Understanding the problem

We need to know the formulas of the surface area and volume of a cube when the edge length is a.
02

Surface Area formula

The surface area (S) of a cube with edge length a can be calculated using the formula: \(S = 6a^2\) (since a cube has 6 equal faces, and each face is a square, so each face has an area of \(a^2\))
03

Volume formula

The volume (V) of a cube with edge length a can be calculated using the formula: \(V = a^3\) (since the volume of a cube is the product of the edge lengths, and all three edge lengths are equal)
04

Calculate the surface area when edge length is increased by a factor of x

When the edge length is increased by a factor of x, the new edge length will be ax. We substitute this new length into the surface area formula: \(S' = 6(ax)^2\) \(S' = 6a^2x^2\) Comparing the original surface area formula (\(S = 6a^2\)) and the new surface area formula (\(S' = 6a^2x^2\)), we can see that the surface area is increased by a factor of \(x^2\) when the edge length is increased by a factor of x.
05

Calculate the volume when edge length is increased by a factor of x

Similar to Step 4, we will now substitute the new edge length (ax) into the volume formula: \(V' = (ax)^3\) \(V' = a^3x^3\) Comparing the original volume formula (\(V = a^3\)) and the new volume formula (\(V' = a^3x^3\)), we can see that the volume increases by a factor of \(x^3\) when the edge length is increased by a factor of x.
06

Conclusion

a) When the edge length of a cube is increased by a factor of x, the surface area is increased by a factor of \(x^2\). b) When the edge length of a cube is increased by a factor of x, the volume is increased by a factor of \(x^3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Area of a Cube
When we think about the surface area of a cube, imagine unwrapping it like a box. You'll find it has six equal square faces. Since each face has an area of \(a^2\) where \(a\) is the length of the edge, the total surface area is calculated as \(S = 6a^2\). This formula is derived from the fact that all six squares are exactly the same in a cube.

But what happens if you change the length of the edge? If the edge length is multiplied by a factor \(x\), the new length becomes \(ax\) and the modified surface area becomes \(S' = 6(ax)^2 = 6a^2x^2\).

  • The surface area increases by a factor of \(x^2\).
This is because both dimensions of each face are scaled by \(x\), leading the entire area to be scaled by \(x^2\).
Volume of a Cube
A cube's volume represents the three-dimensional space it occupies. It's found by taking the edge length \(a\) and using the formula \(V = a^3\). This is because all dimensions of a cube (length, width, height) are equal.

Now, if the edge length increases by a factor \(x\), what happens to the volume? The new edge length is \(ax\), leading to a revised volume of \(V' = (ax)^3 = a^3x^3\).

  • The volume increases by a factor of \(x^3\).
Each dimension of the cube is increased, and in three dimensions, the scaling effects multiply, leading to a larger growth factor in the volume than in the surface area.
Mathematical Formulas
Mathematics provides us with formulas to generalize and simplify the calculation of properties like surface area and volume. For a cube:

  • Surface Area: \(S = 6a^2\)
  • Volume: \(V = a^3\)
These mathematical formulas work on the principle of uniformity in a cube. Every edge and face are equal, which simplifies the calculations immensely. Whether it's determining the material needed to cover a cube (surface area) or finding out how much space it can hold (volume), these formulas are essential tools.

Understanding these formulas is crucial because they help explain geometric properties in a straightforward way, avoiding complex individual calculations.
Scaling in Geometry
Scaling is a fundamental concept in geometry which involves expanding or contracting the size of an object. When the size of an object's dimensions are multiplied by a factor of \(x\), different properties scale differently depending on their nature.

For a cube:

  • When the dimensions of an object are increased by a factor of \(x\), the surface area, being two-dimensional, scales by \(x^2\).
  • In contrast, the volume, being three-dimensional, scales by \(x^3\).
This difference in scaling is a significant concept to grasp, explaining why larger objects grow in capacity more dramatically than just their size alone might suggest. It exemplifies why spatial properties like volume are not linearly related to linear dimensions.

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