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# Joint and Combined Variation

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

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Practice Joint and Combined Variation
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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 $64 \ units^3$ and radius is $4 units$ , what is the height of the cylinder?

### Guidance

The last type of variation is called joint variation . This type of variation involves three variables, usually $x, y$ and $z$ . 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 $\pi$ , so we have $V= \pi r^2h$ . In general, the joint variation equation is $z=kxy$ . Solving for $k$ , we also have $k=\frac{z}{xy}$ .

#### Example A

Write an equation for the given relationships.

a) $y$ varies inversely with the square of $x$ .

b) $z$ varies jointly with $x$ and the square root of $y$ .

c) $z$ varies directly with $x$ and inversely with $y$ .

Solution:

a) $y=\frac{k}{x^2}$

b) $z=kx \sqrt{y}$

c) $z=\frac{kx}{y}$

#### Example B

$z$ varies jointly with $x$ and $y$ . If $x = 3, y = 8,$ and $z = 6$ , find the variation equation. Then, find $z$ when $x = -2$ and $y = 10$ .

Solution: Using the equation when it is solved for $k$ , we have:

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

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

#### Example C

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

Solution: Find the joint variation equation first.

$V=\frac{1}{3} \ Bh$

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

$162&=\frac{1}{3} \cdot 81 \cdot h \\162&=27 \ h \\6&=h$

Intro Problem Revisit The formula for the volume of a cylinder, $V= \pi r^2h$ , is a joint variation equation in which the constant $k = \pi$ .

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

$V = \pi r^2h\\64\pi = \pi 4^2(h)\\64\pi = 16\pi(h)\\4 = h$

Therefore, the cylinder has a height of 4 units.

### Guided Practice

1. Write the equation for $z$ , that varies jointly with $x$ and the cube of $y$ and inversely with the square root of $w$ .

2. $z$ varies jointly with $y$ and $x$ . If $x = 25, z = 10,$ and $k=\frac{1}{5}$ . Find $y$ .

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

a) Write the equation for kinetic energy.

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

1. $z=\frac{kxy^3}{\sqrt{w}}$

2. The equation would be $z=\frac{1}{5}xy$ . Solving for $y$ , we have:

$10&=\frac{1}{5} \cdot 25 \cdot y \\10&=5y \\2&=y$

3. a) $P=\frac{1}{2} \ mv^2$

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

$\frac{104 \ \cancel{km}}{\cancel{hr}} \cdot \frac{\cancel{hr}}{3600 \ s} \cdot \frac{1000 \ m}{\cancel{km}}=0.44 \ \frac{m}{s}$

Now, plug this into the equation from part a.

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

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

### Vocabulary

Joint Variation
Variation where one variable depends upon two independent variables.

### Practice

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

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

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

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

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 $W$ (in joules) done when lifting an object varies jointly with the mass $m$ (in kilograms) of the object and the height $h$ (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 $W, m,$ and $h$ . How much work is done when lifting a 150 kilogram object 2 meters?
4. The intensity $I$ of a sound (in watts per square meter) varies inversely with the square of the distance $d$ (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 $I$ and $d$ . If you are sitting 10 meters back from the stage, what is the intensity of the sound you hear?

### Vocabulary Language: English

Joint Variation

Joint Variation

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