Combined Variation Solutions

• The constant of variation is the value k that relates y    in combination with x and z   .
• 
Combined variation is when one variable depends on two or more others, some    directly    and some     inversely   , all within a single equation.
• 
It is important to    read problems carefully    so that you write equations in the correct form. 
 
•    Joint    variation is a special type of combined variation in which one quantity varies directly with the product of two other quantities, $\underline{ \mathit{y}\mathbf{=}\mathit{k}\mathit{x}\mathit{z}\mathbf{ }\mathbf{ }\mathbf{ }}.$

Example 5

Write the equation in terms of g.

d varies inversely as e, and jointly as f, and the sum of g and h. 

$\begin{array}{rcl}d& =& \frac{kf\left(g+h\right)}{e}\\ \frac{de}{kf}& =& \frac{kf\left(g+h\right)}{kf}\\ \frac{de}{kf}& =& g+h\\ g& =& \frac{de}{kf}–h\end{array}$

Example 6

The volume of a cone varies jointly as the product of the radius squared and the height. Determine the constant of variation of the cone when the volume is 32π cubic meters, the height is $\frac{2}{3}$ meters, and the radius is 12 meters. Then find the volume of a cone when the lengths of the height and radius are switched.

$\begin{array}{rcl}V& =& k{r}^{2}h\\ 32\mathrm{\pi }& =& k{\left(12\right)}^2\left(\frac{2}{3}\right) \\ 32\mathrm{\pi }& =& 144\left(\frac{2}{3}\right)k\\ \frac{32\mathrm{\pi }}{96}& =& \frac{96k}{96}\\ k& =& \frac{\mathrm{\pi }}{3}\end{array}$        $$\begin{gathered} r=\frac{2}{3}, h=12 \\ V=\frac{\mathrm{\pi }}{3}{\left(\frac{2}{3}\right)}^2\left(12\right) \\ V=\frac{\mathrm{\pi }}{\cancel{3}}\left(\frac{4}{9}\right){\left(\cancel{12}\right)}^4 \\ V=\frac{16\mathrm{\pi }}{9} {\mathrm{m}}^3 \end{gathered}$$