Question

The factor of \({\rm{1}}{{\rm{0}}^{{\rm{ - 12}}}}\)in the range of intensities to which the ear can respond, from threshold to that causing damage after brief exposure, is truly remarkable. If you could measure distances over the same range with a single instrument and the smallest distance you could measure was\(1\;{\rm{mm}}\), what would the largest be?

Short answer

The largest distance is \(1{0^9}\;m\).

Step-by-step solution

Given Data

The factor is\(1{0^{ - 12}}\).

The smallest distance is\(d = 1\;mm = 1{0^{ - 3}}\;m\).

Concept

The expression for the factor of intensity is given by,

\(f = \frac{d}{D}\)

Here \(f\) is the factor of intensity, \(d\) is the smallest distance that can be measured, \(D\) is the possible largest distance.

Calculation of the largest distance  

The factor of intensity for the smallest distance\({\rm{d}}\)and the largest distance\({\rm{D}}\)is,

\(\frac{d}{D}\)

Plugging the values,

\(\begin{aligned}{}1{0^{ - 12}} &= \frac{{1{0^{ - 3}}}}{D}\\D &= \frac{{1{0^{ - 3}}}}{{1{0^{ - 12}}}}\\D &= 1{0^9}\;m\end{aligned}\)

Therefore the largest distance is \({10^9}\,{\rm{m}}\).