# Biophysics Problem 39

If a giant species of ant were developed with all body measurements (excluding legs) 10 times as large as the common ant, what would the diameter of the legs, in order to be relatively as strong as the common ant, have to be?

#### First Step

You first need to find out the factor by which the volume increases. Do this now.

#### Variables

The volume would increase by the cube of the linear dimensions. That is, $L^3 = 10^3 = 1000.$ Consequently, the legs would have to be 1000 times as strong as before.

The legs can be considered as pillars (or cylinders), and the strength of a pillar is dependent on its area. To be 1000 times as strong the legs must therefore have 1000 times the area as before.

To achieve this, the diameter of the legs would have to increase by what factor?

#### Calculations

The diameter must increase by the same factor as the radius $(r),$ and recall that area $(A)$ is proportional to the square of the radius; i.e. $A ^{\alpha}r².$

So, taking the square root of 1000, we find the radius needs to increase by 31.7 times. Therefore, the diameter must also increase by 31.7 times.