Question

To visualize the scale of the \(\$ 100\) trillion global economy relative to the value of the earth-conservatively one million times largerlet's think in terms of animal volume. Volume scales like the cube of linear dimension. How much would you have to shrink a dog in linear scale for its volume to be one- millionth its original size? If the typical scale of a dog is \(0.5 \mathrm{~m}\), how large is the shrunken version, and what animal is about this size?

Short answer

Answer: The linear scale factor required is 0.01, resulting in a shrunken dog size of 5mm, which is approximately the size of ants or small beetles.

Step-by-step solution

Determine the relationship between volume scale factor and linear scale factor

We know that the volume scales with the cube of the linear dimension. So, if we have a linear scale factor 'k', then the volume scale factor would be \(k^3\).

Find the linear scale factor corresponding to the given volume scale factor

We are asked to find the linear scale factor when the volume is reduced to one-millionth (i.e., a volume scale factor of \(1 \times 10^{-6}\)) of the original. Since the relationship between the volume and linear scale factors is given by \(k^3\), we can write: \(k^3 = 1 \times 10^{-6}\) To find the linear scale factor 'k', we can take the cube root of both sides: \(k = \sqrt[3]{1 \times 10^{-6}}\)

Calculate the linear scale factor

Computing the cube root, we get: \(k = \sqrt[3]{1 \times 10^{-6}} = 10^{-2}\) So, the linear scale factor is \(0.01\).

Calculate the size of the shrunken dog

We are given that the typical scale of a dog is \(0.5 \mathrm{~m}\). To calculate the size of the shrunken dog, we multiply the original size with the linear scale factor: \(\text{Shrunken dog size} = 0.5 \mathrm{~m} \times 0.01 = 0.005\mathrm{~m}\) The shrunken dog's linear dimension is \(0.005\mathrm{~m}\), which is equal to \(5\mathrm{~mm}\).

Compare the shrunken dog size with known animals

The shrunken dog's size is about \(5\mathrm{~mm}\). This is approximately the size of some insects, like ants or small beetles.

Key concepts

Cube Root

When dealing with the concept of scaling in three dimensions, understanding the cube root is vital. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Written as \( \sqrt[3]{x} \), it answers the question 'What number cubed equals x?'.

For instance, in our exercise, to determine how much we need to shrink the dog to make its volume one-millionth of the original, we used the cube root. Given that the volume scale factor was \( 1 \times 10^{-6} \), we calculated the cube root to find the linear scale factor (the shrinkage in size). This is pivotal because it allowed us to connect the volume reduction directly to the change in the dog's size — indicating that understanding cube roots is essential for grasping volume scaling.

Linear Dimension

Linear dimension refers to a measurement in one direction. This could be the length, width, or height of an object. In our exercise, we discuss scaling an object while maintaining its proportions, meaning that all linear dimensions are uniformly altered.

For the typical dog with an original size of \(0.5\,m\), the linear dimension represents one of these measurements. When we apply the scale factor to the linear dimension, we scale the size of the dog in a single direction. However, since volume is a three-dimensional attribute, this change is expressed to the third power when relating to volume.

Scale Factor

The scale factor is a number which scales, or multiplies, some quantity. In the context of our exercise, we used a scale factor to reduce the size of an object. For volume, which is three-dimensional, we find that when the linear dimensions are scaled by a factor (let's call it \(k\)), the volume is scaled by the cube of that factor, or \(k^3\).

Understanding the scale factor is fundamental because it helps us determine what happens to an object's size (or volume) when it undergoes a proportional change. In our example, we identified a linear scale factor of \(0.01\) — meaning the dog's height, length, and width are all one hundredth of the original size.

Volume and Scaling Relationships

Volume and scaling relationships explore how changing one dimension of an object affects its volume. As volume depends on all three spatial dimensions, it's represented by the formula \( V = l\times w\times h \), where \( l \), \( w \), and \( h \) are the length, width, and height respectively.

In the process of scaling, if we multiply each linear dimension by the scale factor, the new volume is found by multiplying the scale factors of all three dimensions.The result is a simple but powerful rule: scale the linear dimensions by \( k \) and the volume scales by \( k^3 \). This cubic relationship is why a relatively small decrease in linear dimension results in a much greater decrease in volume - a fact clearly illustrated in our canine example.