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Joint and Combined Variation

Identify and solve y=z/kx form equations (three variables)

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Joint Variation

The volume of a cylinder varies jointly with the square of the radius and the height. If the volume of the cylinder is \begin{align*}64 \ \text{units}^3\end{align*} and radius is \begin{align*}4 \text{ units}\end{align*}, what is the height of the cylinder?

Joint Variation

The last type of variation is called joint variation. This type of variation involves three variables, usually \begin{align*}x, y\end{align*} and \begin{align*}z\end{align*}. For example, in geometry, the volume of a cylinder varies jointly with the square of the radius and the height. In this equation the constant of variation is \begin{align*}\pi\end{align*}, so we have \begin{align*}V= \pi r^2h\end{align*}. In general, the joint variation equation is \begin{align*}z=kxy\end{align*}. Solving for \begin{align*}k\end{align*}, we also have \begin{align*}k=\frac{z}{xy}\end{align*}.

Let's write an equation for the following relationships.

  1. \begin{align*}y\end{align*} varies inversely with the square of \begin{align*}x\end{align*}.

\begin{align*}y=\frac{k}{x^2}\end{align*}

  1. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and the square root of \begin{align*}y\end{align*}.

\begin{align*}z=kx \sqrt{y}\end{align*}

  1. \begin{align*}z\end{align*} varies directly with \begin{align*}x\end{align*} and inversely with \begin{align*}y\end{align*}.

\begin{align*}z=\frac{kx}{y}\end{align*}

Now, let's solve the following problems.

  1. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. If \begin{align*}x = 3, y = 8,\end{align*} and \begin{align*}z = 6\end{align*}, find the variation equation. Then, find \begin{align*}z\end{align*} when \begin{align*}x = -2\end{align*} and \begin{align*}y = 10\end{align*}.

Using the equation when it is solved for \begin{align*}k\end{align*}, we have:

\begin{align*}k=\frac{z}{xy}=\frac{6}{3 \cdot 8}=\frac{1}{4}\end{align*}, so the equation is \begin{align*}z=\frac{1}{4}xy\end{align*}.

When \begin{align*}x = -2\end{align*} and \begin{align*}y = 10\end{align*}, then \begin{align*}z=\frac{1}{4} \cdot -2 \cdot 10=-5\end{align*}.

  1. Geometry Connection The volume of a pyramid varies jointly with the area of the base and the height with a constant of variation of \begin{align*}\frac{1}{3}\end{align*}. If the volume is \begin{align*}162 \ units^3\end{align*} and the area of the base is \begin{align*}81 \ units^2\end{align*}, find the height.

Find the joint variation equation first.

\begin{align*}V=\frac{1}{3} \ Bh\end{align*}

Now, substitute in what you know to solve for the height.

\begin{align*}162&=\frac{1}{3} \cdot 81 \cdot h \\ 162&=27 \ h \\ 6&=h\end{align*}

Examples

Example 1

Earlier, you were asked to find the height of the cylinder. 

The formula for the volume of a cylinder, \begin{align*}V= \pi r^2h\end{align*}, is a joint variation equation in which the constant \begin{align*}k = \pi\end{align*}.

We can therefore plug in the given values and solve for h, the height.

\begin{align*}V = \pi r^2h\\ 64\pi = \pi 4^2(h)\\ 64\pi = 16\pi(h)\\ 4 = h\end{align*}

Therefore, the cylinder has a height of 4 units.

Example 2

Write the equation for \begin{align*}z\end{align*}, that varies jointly with \begin{align*}x\end{align*} and the cube of \begin{align*}y\end{align*} and inversely with the square root of \begin{align*}w\end{align*}.

\begin{align*}z=\frac{kxy^3}{\sqrt{w}}\end{align*}

Example 3

\begin{align*}z\end{align*} varies jointly with \begin{align*}y\end{align*} and \begin{align*}x\end{align*}. If \begin{align*}x = 25, z = 10,\end{align*} and \begin{align*}k=\frac{1}{5}\end{align*}, find \begin{align*}y\end{align*}.

The equation would be \begin{align*}z=\frac{1}{5}xy\end{align*}. Solving for \begin{align*}y\end{align*}, we have:

\begin{align*}10&=\frac{1}{5} \cdot 25 \cdot y \\ 10&=5y \\ 2&=y\end{align*}

Example 4

Kinetic energy \begin{align*}P\end{align*} (the energy something possesses due to being in motion) varies jointly with the mass \begin{align*}m\end{align*} (in kilograms) of that object and the square of the velocity \begin{align*}v\end{align*} (in meters per seconds). The constant of variation is \begin{align*}\frac{1}{2}\end{align*}.

Write the equation for kinetic energy.

\begin{align*}P=\frac{1}{2} \ mv^2\end{align*} 

 If a car is travelling 104 km/hr and weighs 8800 kg, what is its kinetic energy?

The second portion of this problem isn’t so easy because we have to convert the km/hr into meters per second.

\begin{align*}\frac{104 \ \cancel{km}}{\cancel{hr}} \cdot \frac{\cancel{hr}}{3600 \ s} \cdot \frac{1000 \ m}{\cancel{km}}=0.44 \ \frac{m}{s}\end{align*}

Now, plug this into the equation from part a.

\begin{align*}P&=\frac{1}{2} \cdot 8800 \ kg \cdot \left(0.44 \ \frac{m}{s}\right)^2 \\ &=1955.56 \ \frac{kg \cdot m^2}{s^2}\end{align*}

Typically, the unit of measurement of kinetic energy is called a joule. A joule is \begin{align*}\frac{kg \cdot m^2}{s^2}\end{align*}.

Review

For questions 1-5, write an equation that represents relationship between the variables.

  1. \begin{align*}w\end{align*} varies inversely with respect to \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
  2. \begin{align*}r\end{align*} varies inversely with the square of \begin{align*}q\end{align*}.
  3. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and \begin{align*}y\end{align*} and inversely with \begin{align*}w\end{align*}.
  4. \begin{align*}a\end{align*} varies directly with \begin{align*}b\end{align*} and inversely with \begin{align*}c\end{align*} and the square root of \begin{align*}d\end{align*}.
  5. \begin{align*}l\end{align*} varies directly with \begin{align*}m\end{align*}, and inversely with \begin{align*}p\end{align*}.

Write the variation equation and answer the given question in each problem.

  1. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. If \begin{align*}x=2,y=3\end{align*} and \begin{align*}z=4\end{align*}, write the variation equation and find \begin{align*}z\end{align*} when \begin{align*}x=-6\end{align*} and \begin{align*}y=2\end{align*}.
  2. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. If \begin{align*}x=5,y=-1\end{align*} and \begin{align*}z=10\end{align*}, write the variation equation and find \begin{align*}z\end{align*} when \begin{align*}x=- \frac{1}{2}\end{align*} and \begin{align*}y=7\end{align*}.
  3. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. If \begin{align*}x=7,y=3\end{align*} and \begin{align*}z=-14\end{align*}, write the variation equation and find \begin{align*}y\end{align*} when \begin{align*}z=-8\end{align*} and \begin{align*}x=3\end{align*}.
  4. \begin{align*}z\end{align*} varies jointly with \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. If \begin{align*}x=8,y=-3\end{align*} and \begin{align*}z=-6\end{align*}, write the variation equation and find \begin{align*}x\end{align*} when \begin{align*}z=12\end{align*} and \begin{align*}y=-16\end{align*}.
  5. \begin{align*}z\end{align*} varies inversely with \begin{align*}x\end{align*} and directly \begin{align*}y\end{align*}. If \begin{align*}x=4,y=48\end{align*} and \begin{align*}z=-2\end{align*}, write the variation equation and find \begin{align*}x\end{align*} when \begin{align*}z=8\end{align*} and \begin{align*}y=96\end{align*}.
  6. \begin{align*}z\end{align*} varies inversely with \begin{align*}x\end{align*} and directly \begin{align*}y\end{align*}. If \begin{align*}x=\frac{1}{2},y=5\end{align*} and \begin{align*}z=20\end{align*}, write the variation equation and find \begin{align*}x\end{align*} when \begin{align*}z=-4\end{align*} and \begin{align*}y=8\end{align*}.

Solve the following word problems using a variation equation.

  1. If 20 volunteers can wash 100 cars in 2.5 hours, find the constant of variation and find out how many cars 30 volunteers can wash in 3 hours.
  2. If 10 students from the environmental club can clean up trash on a 2 mile stretch of road in 1 hour, find the constant of variation and determine how low it will take to clean the same stretch of road if only 8 students show up to help.
  3. The work \begin{align*}W\end{align*} (in joules) done when lifting an object varies jointly with the mass \begin{align*}m\end{align*} (in kilograms) of the object and the height \begin{align*}h\end{align*} (in meters) that the object is lifted. The work done when a 100 kilogram object is lifted 1.5 meters is 1470 joules. Write an equation that relates \begin{align*}W, m,\end{align*} and \begin{align*}h\end{align*}. How much work is done when lifting a 150 kilogram object 2 meters?
  4. The intensity \begin{align*}I\end{align*} of a sound (in watts per square meter) varies inversely with the square of the distance \begin{align*}d\end{align*} (in meters) from the sound’s source. At a distance of 1.5 meters from the stage, the intensity of the sound at a rock concert is about 9 watts per square meter. Write an equation relating \begin{align*}I\end{align*} and \begin{align*}d\end{align*}. If you are sitting 10 meters back from the stage, what is the intensity of the sound you hear?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 9.3. 

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Vocabulary

Joint Variation

Variables exhibit joint variation if one variable varies directly as the product of two or more other variables.

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