Equations to help: $V = \frac 43\pi r^3$, $p = m/v$ , $s = ut + \frac12 at^2$ , $v^2 = u^2 + 2as$
In each this situation you are told which data is collected. Please determine using the collected formulas in the space below how those variables should be modified in order to create a graph with a liner relationship.
- Data collected: Volume and Radius
- What equation could be used to relate these variables?
- What should be plotted on the y and x axis to create a linear relationship?
- Determine an equation in variable form for the slope of the line.
I don't understand how I am supposed to linearize volume and radius...
The first equation is the volume of a sphere as a function of radius. There are two ways to draw a straight line.
Option 1, plot the volume as a function of $r^3$. The slope of the line will be $\frac 43 \pi$, and the intercept should be $0$.
Option 2, take the logarithm of both sides of the first equation: $$\ln V=\ln\left(\frac 43 \pi r^3\right)=\ln\left(\frac 43 \pi \right)+3\ln r$$ So if you plot $\ln V$ as a function of $\ln r$, you get a line with the slope $3$ and intercept $\ln\left(\frac 43 \pi \right)$