Skip to main content

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 \(~231.90=18.85r^2~\text{.}\)

Solution

The equation is quadratic. We solve by extraction of roots.

\begin{align*} 231.90 \amp = 18.85 r^2 \amp\amp \blert{\text{Isolate the squared expression.}}\\ 12.302 \amp = r^2 \amp\amp \blert{\text{Take square roots.}}\\ r \amp = 35 \end{align*}
Example 3.2.

Solve \(~2.8125 = \dfrac{36}{n}\text{.}\)

Solution

We must first get the variable out of the denominator.

\begin{align*} n(2.8125) \amp = \dfrac{36}{n} n \amp\amp \blert{\text{Multiply boyh sides by}~n.}\\ 2.8125n \amp = 36 \amp\amp \blert{\text{Divide both sides by 2.8125.}}\\ n \amp = 12.8 \end{align*}
Example 3.3.

Solve \(~0.5547 = \dfrac{1500}{d^2}\text{.}\)

Solution

We must first get the variable out of the denominator.

\begin{align*} d^2(0.5547) \amp = \dfrac{1500}{d^2} d^2 \amp\amp \blert{\text{Multiply boyh sides by}~d^2.}\\ 0.5547d^2 \amp = 1500 \amp\amp \blert{\text{Divide both sides by 0.5547.}}\\ d^2 \amp = 2704.16 \amp\amp \blert{\text{Take square roots.}}\\ d \amp = 52 \end{align*}

Subsubsection Exercises

Solve \(~1371.8=25R^3\text{.}\)

Answer
\(3.8\)

Solve \(~13.03=\dfrac{380}{h^2}\text{.}\)

Answer
\(5.4\)

Solve \(~0.065=\dfrac{12}{p}\text{.}\)

Answer
\(184.6\)

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 \(V=0.2s^3\text{.}\)

Solution

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.

\begin{align*} \text{If } s=1, \amp V = 0.2(1)^3 = 0.2\\ \text{If } s=2, \amp V = 0.2(2)^3 = 1.6\\ \text{If } s=3, \amp V = 0.2(3)^3 = 5.4 \end{align*}

The graph is shown below.

cubic
Example 3.8.

Sketch a graph of \(H=\dfrac{48}{w}\text{.}\)

Solution

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.

\begin{align*} \text{If } w=2, \amp H = \dfrac{48}{2} = 24\\ \text{If } w=6, \amp H = \dfrac{48}{6} = 8\\ \text{If } w=12, \amp H = \dfrac{48}{12} = 4 \end{align*}

The graph is shown below.

cubic

Subsubsection Exercises

Plot three points and sketch a graph of \(B=\dfrac{0.8}{d^2}\text{.}\)

reciprocal squared
Answer
\((1,0.8),~(2,0.2),~(4,0.05)\)
reciprocal squared

Plot three points and sketch a graph of \(d=\dfrac{3}{8}t^2\text{.}\)

grid
Answer
\(\left(1,\dfrac{3}{8}\right)\text{,}\) \(\left(2,\dfrac{3}{2}\right)\text{,}\) \((4,6)\)
parabola

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 \(x=2.5\text{.}\)

Solution

Because \(y\) varies directly with the square of \(x\text{,}\) we know that \(y=kx^2\text{.}\) We substitute the given values to find

\begin{align*} 100 \amp = k(2.5)^2 \amp\amp \blert{\text{Solve for}~k.}\\ k \amp = \dfrac{100}{2.5^2} = 16 \end{align*}

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 \(x=0.16\text{.}\)

Solution

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

\begin{align*} 4687.5 \amp = \dfrac{k}{0.16^2} \amp\amp \blert{\text{Solve for}~k.}\\ k \amp = 4687.5(0.16)^2=120 \end{align*}

The constant of variation is 120, and the variation equation is \(~y=\dfrac{120}{x^2}\text{.}\)

Subsubsection Exercises

Find the constant of variation and the variation equation:

\(y\) varies inversely with \(x\text{,}\) and \(y=31.25\) when \(x=640\text{.}\)

Answer
\(k=20,000\) and \(y=\dfrac{20,000}{x}\)

Find the constant of variation and the variation equation:

\(y\) varies directly with the cube of \(x\text{,}\) and \(y=119,164\) when \(x=6.2\text{.}\)

Answer
\(k=500\) and \(y=500x^3\)