, in political science, observes that in many countries, the number of seats in the legislative body most similar to the US House of Representatives is approximately modeled by the formula \(s = \sqrt[3]{P}\text{,}\) where \(s\) is the number of seats in the legislative body and \(P\) is the country’s population.
The cube root law is a power model, even though it doesn’t look like one at first. Use the fact that cube roots are the same as the 1/3rd power to rewrite the cube root law so that it looks like a power model.
The population of the United States was about 332 million in 2021. Write 332 million in scientific notation, then use the cube root law to predict the size of the US House of Representatives.
The US House of Representatives hasn’t changed size since it was set to 435 by the Apportionment Act of 1911. The US population in 1911 was about 94 million. Did the cube root law give a good estimate in 1911? Don’t forget to rewrite 94 million in scientific notation first!
Police use the formula \(v = \sqrt{20L}\) to estimate the speed of a car, \(v\text{,}\) in miles per hour, based on the length, \(L\text{,}\) in feet, of its skid marks when suddenly braking on a dry, asphalt road.
The United States had a national speed limit of 55 mph from 1974 - 1987, and speed limits of 55 mph were still common around the country through the late 1990s. How long would the skid marks be of a car suddenly braking after going 55 mph on a dry, asphalt road?
The fastest posted speed limit in the US is 85 mph. How long would the skid marks be of a car suddenly braking after going 85 mph on a dry, asphalt road?
The resting energy expenditure (REE) is the amount of energy a person or animal needs for basic bodily functions. In the 1930s, biologist Max Kleiber 2
Z. Wang, T. O’Connor, S. Heshka, and s. Heymsfield. ``The Reconstruction of Kleiber’s Law at the Organ-Tissue Level.’’ The Journal of Nutrition v. 131, no. 11. doi: 10.1093/jn/131.11.2967
discovered that there is a relationship between the REE of an adult mammal and its body mass:
\begin{equation*}
R = 70 \times M^{3/4}
\end{equation*}
where \(R\) is a mammal’s REE (in kcal/d) and \(M\) is its body mass (in kg).
In order to find the body mass of a mammal with REE 1400 kcal/d, we need to undo the 3/4ths power. We don’t usually talk about the 3/4ths root. Instead, we stick with fractional exponents. To undo the 3/4ths power, we’ll use the 1/(3/4)ths power. Here’s how this example works out!
Note: 1/(3/4) is hard to read and work with! It’s important to get the parentheses right, since 1/(3/4) and (1/3)/4 are different (try to plug both into a calculator and see for yourself!). Instead of doing 1/(3/4), it’s more common to either turn 3/4 into a decimal and do 1/0.75, or to use fraction arithmetic rules to rewrite 1/(3/4) as 4/3.
where \(a\) is the planet’s distance from the Sun in AU and \(T\) is the number of days the planet takes to orbit the Sun. Note that 1 AU is the distance between Earth and the Sun.
Kepler’s Third Law is a power model in disguise! Talk to your table and decide how to use fractional exponents to rewrite your equation from the previous part with a fractional exponent.
Use Kepler’s Third Law to predict the distance from the Sun to a planet with a 10,776 day year. (What planet is this? Use the Internet to find out! How good was your distance estimate?)
