Question

The frequencies to which the ear responds vary by a factor of \[{\rm{1}}{{\rm{0}}^{\rm{3}}}\]. Suppose the speedometer on your car measured speeds differing by the same factor of \[{\rm{1}}{{\rm{0}}^{\rm{3}}}\], and the greatest speed it reads is \[{\rm{90}}{\rm{.0}}\;{\rm{mi/h}}\]. What would be the slowest nonzero speed it could read?

Short answer

The smallest speed is \[90 \times 1{0^{ - 3}}\;mi/h\].

Step-by-step solution

Given Data

The factor is\[1{0^3}\].

The greatest speed is \[V = 90.0\;mi/h\].

Concept

The expression for the factor of intensity is given by,

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

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

Calculation of the largest distance  

The factor of intensity for the smallest speed\[{\rm{v}}\]and the greatest speed\[{\rm{V}}\]is,

\[f = \frac{v}{V}\]

Plugging the values,

\[\begin{aligned}{c}1{0^3} &= \frac{{90}}{V}\\V &= \frac{{90}}{{1{0^3}}}\\V &= 90 \times 1{0^{ - 3}}\;mi/h\end{aligned}\]

Therefore the smallest speed is \[90 \times 1{0^{ - 3}}\;mi/h\].