Variations

Frank Ma © 2011

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant.

Variations

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

Example A. Translate the following phrases into equations.

a. y varies directly to x.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

y = kxz for some k.

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

y = kxz for some k.

c. y varies directly to x2z2

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

y = kxz for some k.

c. y varies directly to x2z2

y = kx2z2 for some k.

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

y = kxz for some k.

c. y varies directly to x2z2

y = kx2z2 for some k.

d. The cost C varies directly with the square of the length L.

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

y = kxz for some k.

c. y varies directly to x2z2

y = kx2z2 for some k.

d. The cost C varies directly with the square of the length L.

The square of the length L is L2.

Example A. Translate the following phrases into equations.

a. y varies directly to x. y = kx for some k.

We say the variable y varies (directly) to an expression f if y = k·fwhere k is a constant. The formula y = k* f is called the general (direct) variation equation.

Variations

b. y varies directly to xz

y = kxz for some k.

c. y varies directly to x2z2

y = kx2z2 for some k.

d. The cost C varies directly with the square of the length L.

The square of the length L is L2. Hence the general equation is y = kL2 for some k.

We say the variable y varies inversely to an expression f if

y =

where k is a constant.

Variations

kf

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

Example B. Translate the following phrases into equations.

a. y varies inversely to x.

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

Example B. Translate the following phrases into equations.

a. y varies inversely to x.kx

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

y = where k is a constant

Example B. Translate the following phrases into equations.

a. y varies inversely to x.

kx

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

b. y varies inversely to x2z

y = where k is a constant

Example B. Translate the following phrases into equations.

a. y varies inversely to x.

kx

kx2z

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

b. y varies inversely to x2z

y = where k is a constant

y = where k is a constant

Example B. Translate the following phrases into equations.

a. y varies inversely to x.

kx

kx2z

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

b. y varies inversely to x2z

y = where k is a constant

y = where k is a constant

c. The intensity of light I varies inversely to the square of distance D

Example B. Translate the following phrases into equations.

a. y varies inversely to x.

kx

kx2z

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

b. y varies inversely to x2z

y = where k is a constant

y = where k is a constant

c. The intensity of light I varies inversely to the square of distance D

The square of distance D is D2.

Example B. Translate the following phrases into equations.

a. y varies inversely to x.

kx

kx2z

We say the variable y varies inversely to an expression f if

y =

where k is a constant. The formula y = is called the general inverse variation equation.

Variations

kf k

f

b. y varies inversely to x2z

y = where k is a constant

y = where k is a constant

c. The intensity of light I varies inversely to the square of distance D

The square of distance D is D2.

Hence I = kD2 where k is a constant.

In general, a variation problem gives the type of the variation and the values of the variables.

Variations

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Since y varies directly to x, the general equation is y = kx.

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Since y varies directly to x, the general equation is y = kx. Set y = –4 and x = –6 in the general equation.

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Since y varies directly to x, the general equation is y = kx. Set y = –4 and x = –6 in the general equation.

–6 = k(–4)

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

k = =–6 –4 2

3

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Since y varies directly to x, the general equation is y = kx. Set y = –4 and x = –6 in the general equation.

–6 = k(–4) so

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

k = =–6 –4

If we put this into the general equation, we have the specific equation:

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Since y varies directly to x, the general equation is y = kx. Set y = –4 and x = –6 in the general equation.

–6 = k(–4) so

23

Example C. a. Given that y varies directly to x and y = –4 when x = –6. Find the constant k and the specific variation equation.

k = =–6 –4

If we put this into the general equation, we have the specific equation:

In general, a variation problem gives the type of the variation and the values of the variables. From these, we solve for the constant k and find the specific (exact) variation equation.

Variations

Since y varies directly to x, the general equation is y = kx. Set y = –4 and x = –6 in the general equation.

–6 = k(–4) so

y = x

23

23

Variationsb. Find the value of x if y = 20.

Variationsb. Find the value of x if y = 20.

20 = x23

Substitute y = 20 into the specific equation y = x23

Variationsb. Find the value of x if y = 20.

20 = x23

Substitute y = 20 into the specific equation

20 * = x23

y = x23

b. Find the value of x if y = 20.

Variations

20 = x23

Substitute y = 20 into the specific equation

20 * = x23

So x = 40/3 if y = 20.

y = x23

b. Find the value of x if y = 20.

Variations

20 = x23

Substitute y = 20 into the specific equation

20 * = x23

So x = 40/3 if y = 20.In application, the language of variation sums up the basic relation of two or more types of variables.

y = x23

b. Find the value of x if y = 20.

Variations

20 = x23

Substitute y = 20 into the specific equation

20 * = x23

So x = 40/3 if y = 20.In application, the language of variation sums up the basic relation of two or more types of variables.

y = x23

Example D. The weight W of an object varies inversely to the square of the distance D from the center of the earth. A person weighs 160 pounds on the surface which is 4000 miles to the center of the earth. What would the person weigh if he’s 6000 miles above the surface of the earth?

b. Find the value of x if y = 20.

Variations

20 = x23

Substitute y = 20 into the specific equation

20 * = x23

So x = 40/3 if y = 20.In application, the language of variation sums up the basic relation of two or more types of variables.

y = x23

4000 m.

W = 160

Example D. The weight W of an object varies inversely to the square of the distance D from the center of the earth. A person weighs 160 pounds on the surface which is 4000 miles to the center of the earth. What would the person weigh if he’s 6000 miles above the surface of the earth?

b. Find the value of x if y = 20.

Variations

20 = x23

Substitute y = 20 into the specific equation

20 * = x23

So x = 40/3 if y = 20.In application, the language of variation sums up the basic relation of two or more types of variables.

y = x23

4000 m.

6000 m.

W = 160

W = ?Example D. The weight W of an object varies inversely to the square of the distance D from the center of the earth. A person weighs 160 pounds on the surface which is 4000 miles to the center of the earth. What would the person weigh if he’s 6000 miles above the surface of the earth?

VariationsThe square of the distance D is D2.

VariationsThe square of the distance D is D2.

So the general equation is W = . D2k

VariationsThe square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

Substitute W = 160, D = 4000 into the specific equation.

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

k160 * 16,000,000 =

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

k160 * 16,000,000 =

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth.

k160 * 16,000,000 =

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)

k160 * 16,000,000 =

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)Hence the specific equation is

k160 * 16,000,000 =

W = D22.56 * 109

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)Hence the specific equation is

When the person is 6000 miles above the surface D = 10000,

k160 * 16,000,000 =

W = D22.56 * 109

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)Hence the specific equation is

When the person is 6000 miles above the surface D = 10000,

we have

k160 * 16,000,000 =

W = D22.56 * 109

W = (10000)22.56 * 109

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)Hence the specific equation is

When the person is 6000 miles above the surface D = 10000,

we have

k160 * 16,000,000 =

W = D22.56 * 109

W = (10000)22.56 * 109

= 1082.56 * 109

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)Hence the specific equation is

When the person is 6000 miles above the surface D = 10000,

we have

k160 * 16,000,000 =

W = D22.56 * 109

W = (10000)22.56 * 109

= 1082.56 * 109

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

108

10

Variations

160 = (4000)2

kSubstitute W = 160, D = 4000 into the specific equation.

So k = 2.56 * 109

(This constant is specific to earth. For another planet, the general equation is the same but k would be different.)Hence the specific equation is

When the person is 6000 miles above the surface D = 10000,

we have

k160 * 16,000,000 =

W = D22.56 * 109

W = (10000)22.56 * 109

= 1082.56 * 109

The square of the distance D is D2.

So the general equation is W = . D2k We are to find k .

108

10

= 25.6 lb

VariationsIf the expression f is a product of two or more variables, we also say y varies jointly to these variables.

VariationsIf the expression f is a product of two or more variables, we also say y varies jointly to these variables. Hence

“y varies directly to xz” is the same as “y varies jointly to x and z”.

VariationsIf the expression f is a product of two or more variables, we also say y varies jointly to these variables. Hence

“y varies directly to xz” is the same as “y varies jointly to x and z”.

“y varies directly to x2z2 ” is the same as “y varies jointly to x2 and z2”.

VariationsExercise A. Write down the general equations for the following variation statements.

5. T varies directly to P. 6. T varies inversely to V.

1. R varies directly to D. 2. R varies inversely to T.

7. A varies directly to r2.9. C varies directly to D3

8. y varies inversely to d2.

3. S varies directly to T. 4. S varies inversely to N.

10. C varies directly to xy.11. The rate R that a car is traveling at varies directly to the distance D it covers in a fixed amount of time.12. The rate R that a car is traveling at varies inversely to the time T it takes to travel a fixed distance.

Each person in a group of N people has to chip in to buy a large pizza that cost $T. Let S be the share of each person, then13. S varies directly to T 14. N varies inversely to S.

Variations

16. The temperature varies inversely to the volume.

17. The area of a circle varies directly to the square of its radius.18. The light–intensity varies inversely to the square of the distance.19. The mass required varies inversely to the square of the speed it was traveling.

20. The cost of cheese balls varies directly to the 3rd power (cube) of the diameter of the ball.

22. The time seemly has passed doing something varies inversely to how much fun there was doing it.

21. The fun of doing something varies inversely to how much time seemly has passed doing it.

15. The temperature varies directly to the pressure.

Assign variables and write down the variation equations.

Variations

24. The light–intensity L underwater varies inversely to the square of the depth of the distance. If at 3 feet the intensity L is 5 ft–candle, find the specific equation and what is the intensity at the depth of 10 ft.25. The price of a pizza varies directly to the square of the diameter of the pizza. Given that a 6” pizza is $5.50, find the specific equation for the price in terms of the diameter. What should be the price of a 12” pizza? 26. The cost of a solid chocolate ball varies directly to the cube of its diameter. A chocolate ball with 3” diameter cost $25, find the specific equation of the cost in terms of the diameter and how much is one with a 1– foot diameter?

23. The number of marbles N that we are able to buy varies inversely to the price P of a marble. We’re able to buy 30 marbles at $45 each. What is the specific equation of the N in terms of P and what is the price P if we are only able to buy 10 marbles?