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Section 3.5 Joint Variation

Subsection 1. Evaluate a function of two variables

We evaluate a function of two variables just as we do any other function: by substituting the given values for the variables.

Subsubsection Examples

Example 3.61.

Evaluate \(~f(x,y) = 1.6x^2 + 2.4y~\) for \(x=24\) and \(y=300\text{.}\)

Solution

We substitute \(\alert{240}\) for \(x\) and \(\alert{300}\) for \(y\text{.}\)

\begin{align*} f(\alert{24}, \alert{300}) \amp = 1.6(\alert{24}^2) + 2.4(\alert{300})\\ \amp = 1.6(576) + 2.4(300) = 921.6 + 720 = 1641.6 \end{align*}
Example 3.62.

\(h(s,t) = 12s^{2/3}t^{1/4}\text{.}\) Evaluate \(h(35,60).\) Round your answer to thousandths.

Solution

We substitute \(\alert{35}\) for \(s\) and \(\alert{60}\) for \(t\text{.}\)

\begin{align*} h(\alert{35}, \alert{60}) \amp = 12(\alert{35}^{2/3})(\alert{60}^{1/4})\\ \amp = 12(10.700)(2.783) = 357.353 \end{align*}

Subsubsection Exercises

Evaluate \(~g(a,b) = \dfrac{4a^3}{b^{1/2}}~\) for \(a=8\) and \(b=12\text{.}\)

Answer
\(591.207\)

\(~F(d,w) = 6.5d^{0.25}w^{0.4}~\text{.}\) Evaluate \(~F(32, 18)\text{.}\)

Answer
\(49.126\)

Subsection 2. Read a table for a function of two variables

Values of the first input variable are listed in the first column, and values of the second input variable are listed in the first row. The output values are shown in the body of the table.

Subsubsection Example

Example 3.65.

The table shows values for \(~z=f(x,y)\text{.}\)

\(x~{\Large\setminus}~y\) \(1\) \(2\) \(3\) \(4\) \(5\)
\(1\) \(1\) \(4\) \(9\) \(16\) \(25\)
\(2\) \(2\) \(8\) \(18\) \(32\) \(50\)
\(3\) \(3\) \(12\) \(27\) \(48\) \(75\)
\(4\) \(4\) \(16\) \(36\) \(64\) \(100\)
\(5\) \(5\) \(20\) \(45\) \(80\) \(125\)

  1. Evaluate \(~f(3,5)\text{.}\)
  2. Solve the equation \(~f(x,y) = 16.~\) Give your answer as an ordered pair.
  3. Is the function \(~f(2,y)~\) increasing or decreasing?
Solution
  1. The inputs are \(x=3\) and \(y=5\text{.}\) We look in the third row and fifth column to find \(~f(3,5) = 75.\)
  2. We see the entry 16 in the fourth row and the second column, so \(x=4\) and \(y=2\text{.}\) The solution is \((4,2).\)
  3. The entries in the second row are increasing as \(y\) increases, so \(~f(2,y)~\) is increasing.

Subsubsection Exercise

The table shows values for \(~z=f(x,y)\text{.}\)

\(x~{\Large\setminus}~y\) \(10\) \(15\) \(20\) \(25\) \(30\)
\(4\) \(82\) \(61\) \(58\) \(30\) \(26\)
\(8\) \(78\) \(59\) \(60\) \(35\) \(26\)
\(12\) \(75\) \(53\) \(62\) \(40\) \(26\)
\(16\) \(67\) \(46\) \(64\) \(45\) \(26\)
\(20\) \(62\) \(40\) \(66\) \(50\) \(26\)

  1. Evaluate \(~f(8,20)\text{.}\)
  2. Solve the equation \(~f(x,y) = 62.~\) Give your answer as an ordered pair.
  3. For which value(s) of \(x\) is the function \(~f(x,y)~\) decreasing?
Answer
  1. \(\displaystyle 60\)
  2. \((12,20)\) and \((20,10)\)
  3. \(\displaystyle x=4\)

Subsection 3. Find a constant of variation

Subsubsection Example

Example 3.67.

The table shows values for \(~z=f(x,y)\text{,}\) where \(z\) varies directly with \(x\) and inversely with the square of \(y\text{.}\) Find the constant of variation.

\(x~{\Large\setminus}~y\) \(1\) \(2\) \(3\) \(4\)
\(1\) \(60\) \(15\) \(\dfrac{20}{3}\) \(\dfrac{15}{4}\)
\(2\) \(120\) \(30\) \(\dfrac{40}{3}\) \(\dfrac{15}{2}\)
\(3\) \(180\) \(45\) \(20\) \(\dfrac{45}{2}\)
\(4\) \(240\) \(60\) \(\dfrac{80}{3}\) \(15\)

Solution

The function \(f\) has the form \(z=\dfrac{kx}{y^2}\text{.}\) To find \(k\) we substitue values for the variables. For example, \(x=60\) when \(x=1\) and \(y=1\text{,}\) so

\begin{equation*} 60=\dfrac{k(1)}{1^2} \end{equation*}

and \(k=60\text{.}\) Thus, \(z=\dfrac{60x}{y^2}\text{.}\)

Subsubsection Exercise

The table shows values for \(~z=f(x,y)\text{,}\) where \(z\) varies directly with \(x^2\) and \(y^2\text{.}\) Find the constant of variation.

\(x~{\Large\setminus}~y\) \(2\) \(4\) \(6\) \(8\)
\(4\) \(16\) \(64\) \(144\) \(256\)
\(8\) \(64\) \(256\) \(576\) \(1024\)
\(10\) \(100\) \(400\) \(900\) \(1600\)
\(12\) \(144\) \(576\) \(1296\) \(2304\)

Answer
\(z=\dfrac{1}{4}x^2y^2\)