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Joint and Combined Variation

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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?

Answers

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?

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