Section 3.1 Variation
Subsection 1. Solve a variation equation
We have encountered equations of this form before. Here is a quick review.
Subsubsection Examples
In these examples, we assume all variables are positive. We round answers to tenths.
Example 3.1.
Solve \(~231.90=18.85r^2~\text{.}\)
The equation is quadratic. We solve by extraction of roots.
Example 3.2.
Solve \(~2.8125 = \dfrac{36}{n}\text{.}\)
We must first get the variable out of the denominator.
Example 3.3.
Solve \(~0.5547 = \dfrac{1500}{d^2}\text{.}\)
We must first get the variable out of the denominator.
Subsubsection Exercises
Checkpoint 3.4.
Solve \(~1371.8=25R^3\text{.}\)
Checkpoint 3.5.
Solve \(~13.03=\dfrac{380}{h^2}\text{.}\)
Checkpoint 3.6.
Solve \(~0.065=\dfrac{12}{p}\text{.}\)
Subsection 2. Sketch a variation graph
The graphs of variations are transformations of the basic graphs \(y=x^n\) and \(y=\dfrac{1}{x^n}\text{.}\)
Subsubsection Examples
Example 3.7.
Sketch a graph of \(V=0.2s^3\text{.}\)
We know that the graph has the shape of the basic function \(y=x^3\text{,}\) so all we need are a few points to "anchor" the graph.
The graph is shown below.
Example 3.8.
Sketch a graph of \(H=\dfrac{48}{w}\text{.}\)
We know that the graph has the shape of the basic function \(y=\dfrac{1}{x}\text{,}\) so all we need are a few points to "anchor" the graph.
The graph is shown below.
Subsubsection Exercises
Checkpoint 3.9.
Plot three points and sketch a graph of \(B=\dfrac{0.8}{d^2}\text{.}\)
Checkpoint 3.10.
Plot three points and sketch a graph of \(d=\dfrac{3}{8}t^2\text{.}\)
Subsection 3. Find the constant of variation
If we know the type of variation and the coordinates of one point ont the graph, we can find the variation equation.
Subsubsection Examples
Example 3.11.
Find the constant of variation and the variation equation:
\(~~~y\) varies directly with the square of \(x\text{,}\) and \(y=100\) when \(x=2.5\text{.}\)
Because \(y\) varies directly with the square of \(x\text{,}\) we know that \(y=kx^2\text{.}\) We substitute the given values to find
The constant of variation is 16, and the variation equation is \(~y=16x^2\text{.}\)
Example 3.12.
Find the constant of variation and the variation equation:
\(~~~y\) varies inversely with the square of \(x\text{,}\) and \(y=4687.5\) when \(x=0.16\text{.}\)
Because \(y\) varies inversely with the square of \(x\text{,}\) we know that \(y=\dfrac{k}{x^2}\text{.}\) We substitute the given values to find
The constant of variation is 120, and the variation equation is \(~y=\dfrac{120}{x^2}\text{.}\)
Subsubsection Exercises
Checkpoint 3.13.
Find the constant of variation and the variation equation:
\(y\) varies inversely with \(x\text{,}\) and \(y=31.25\) when \(x=640\text{.}\)
Checkpoint 3.14.
Find the constant of variation and the variation equation:
\(y\) varies directly with the cube of \(x\text{,}\) and \(y=119,164\) when \(x=6.2\text{.}\)