Power models with negative powers are often expressed in words using the language of inverse proportionality, or by saying that one quantity varies indirectly with another. For example:
The force \(F\) (in pounds) needed on a wrench handle to loosen a certain bolt varies inversely with the length \(L\) of the handle. So
\begin{equation*}
F = \frac{c}{L} \qquad \text{or} \qquad F = cL^{-1}
\end{equation*}
The weight, \(w\text{,}\) of an object above the surface of Earth varies inversely with the square of distance from the center of Earth, \(d\text{.}\) So
\begin{equation*}
w = \frac{c}{d^2} \qquad \text{or} \qquad w = cd^{-2}
\end{equation*}
In each of the following situations, set up the power model being described. Use the information given to find the coefficient, then answer the question.
The loudness, \(L\text{,}\) of a sound (measured in decibels, dB) is inversely proportional to the square of the distance, \(d\text{,}\) from the source of the sound. When a person 7.0 feet from a jetski, it is 75.0 decibels loud. How loud is the jetski when the person is 45 feet away?
The number of hours, \(h\text{,}\) required to build a fence is inversely proportional to the number of people, \(n\) working on the fence. If it takes 4 people 31 hours to complete the fence, then how long will it take 11 people to build the fence?
The two octaves around middle A on a piano have frequencies of about 220 - 880 Hz. Sketch a graph showing the relationship of power to frequency in pink noise for the two octaves around middle A. Hint: You don’t need to know anything about music to solve this problem. The instructions are telling you that the horizontal axis should start at 220 and go up to 880. Choose some numbers between 220 and 880, plug them into the formula to find the power, and then use these points to make a graph.
The model for pink noise is also a power model! We can rewrite it as
\begin{equation*}
P = f^{-1}\text{.}
\end{equation*}
Talk to your table and brainstorm reasons why it might make sense for negative exponents to be related to variables in the denominator of fractions. Hint: You thought about this in the prep activities.
Sketch a graph showing the relationship of power to frequency in brown noise for the two octaves around middle A. Hint: What steps did you use to sketch a graph for pink noise?
Walking on snow is difficult because our feet tend to puncture the snow (high pressure applied to low strength snow). We can use snowshoes to avoid this problem: they reduce pressure with minimal increase in force (weight).
Without snowshoes, each of Guido’s feet has an area of \(22\text{ in}^2\text{.}\) What is the pressure he exerts on the snow? Hint: Guido has two feet! So, what is the total area of his feet?
If Guido wants to exert a pressure of \(1 \text{ lb/in}^2\text{,}\) how big do his snowshoes need to be? If he wants to exert a pressure of \(0.5 \text{ lb/in}^2\)
Let’s stretch our understanding of negative exponents and logarithms! Evaluate each of the logs below without using a calculator. You can check your answer with a calculator, but should first find them without one.
