Problem 40 - College Physics

Question

In the redesign of a machine, a metal cubical part has each of its dimensions tripled. By what factor do its surface area and volume change?

Step-by-Step Solution

Answer

Surface area increases by a factor of 9, and volume increases by a factor of 27.

Step 1: Understanding the Problem

We need to determine how the surface area and volume of a cubical part change when each of its dimensions is tripled.

Step 2: Original Surface Area Calculation

The formula for the surface area of a cube with side length \( s \) is \( 6s^2 \). This represents the total area of all six faces of the cube.

Step 3: New Surface Area Calculation

If each dimension is tripled, the new side length is \( 3s \). The surface area of the enlarged cube is \( 6(3s)^2 = 6 \times 9s^2 = 54s^2 \).

Step 4: Surface Area Factor

To find the factor change in surface area, divide the new surface area by the original surface area: \( \frac{54s^2}{6s^2} = 9 \). The surface area increases by a factor of 9.

Step 5: Original Volume Calculation

The formula for the volume of a cube with side length \( s \) is \( s^3 \). This represents the space taken up by the cube.

Step 6: New Volume Calculation

With the side length tripled to \( 3s \), the new volume is \((3s)^3 = 27s^3 \).

Step 7: Volume Factor

To find the factor change in volume, divide the new volume by the original volume: \( \frac{27s^3}{s^3} = 27 \). The volume increases by a factor of 27.

Key Concepts

Geometry

Geometry is the branch of mathematics that deals with the size, shape, and properties of figures and spaces. It helps us understand how objects behave in space and allows us to calculate critical attributes like area and volume. Understanding geometry is essential in everyday life, from architecture to design, and especially when working with shapes like cubes. When we analyze a cube, we focus on its shape—a solid with six equal square faces. Knowing the basics of geometric shapes, including cubes, aids in exploring concepts such as scaling and its effect on surface area and volume.

Scale Factor

The scale factor is a crucial concept in geometry, especially when resizing objects. It tells us how much larger or smaller an object becomes when its dimensions are altered. In this exercise, we triple the dimensions of a cubical part, meaning the scale factor is 3.

• A scale factor greater than 1 indicates an enlargement.
• A scale factor less than 1 indicates a reduction.

When applying a scale factor to a cube, every dimension is multiplied by this factor. This resizing impacts both surface area and volume, effectively determining the extent to which these properties increase or decrease.

Cube Dimensions

In geometry, a cube is defined by its dimensions—specifically, its side length. Every side length of a cube is equal, which simplifies calculations for properties like surface area and volume. When altering cube dimensions, such as tripling each side, we analyze how the entire object's size changes while maintaining its geometric properties.

• Originally, each side of the cube is length \( s \).
• Triple each side results in a new length of \( 3s \).

Altering dimensions like this directly influences other properties, which can be systematically calculated using standard formulas.

Surface Area Formula

The surface area of a cube is calculated by considering all six faces. The formula is given by: \[ 6s^2 \]where \( s \) is the original side length. Upon tripling the side length to \( 3s \), the new surface area becomes:\[ 6(3s)^2 = 6 \times 9s^2 = 54s^2 \]To find how the surface area changes, we compare the new surface area to the original:\[ \frac{54s^2}{6s^2} = 9 \]This shows an increase by a factor of 9. This example illustrates that surface area changes with the square of the scale factor.

Volume Formula

The volume of a cube initially depends on its side length, calculated as:\[ s^3 \]Upon changing each side length to \( 3s \), the new volume formula is:\[ (3s)^3 = 27s^3 \]To determine the factor by which the volume increases, compare the new volume to the original:\[ \frac{27s^3}{s^3} = 27 \]This means the volume increases by a factor of 27, illustrating that volume changes with the cube of the scale factor. When you resize an object, its volume responds much more dramatically than its surface area, which is important in many practical applications.