Mathematical Modeling and Variationbanach.millersville.edu/~bob/math160/Modeling/main.pdf · The frequency of vibration of a piano string varies directly as ... Robert Buchanan Mathematical

Mathematical Modeling and VariationMATH 160, Precalculus

J. Robert Buchanan

Department of Mathematics

Fall 2011

J. Robert Buchanan Mathematical Modeling and Variation

Objectives

In this lesson we will learn to:write mathematical models for direct variation,write mathematical models for direct variation as an nthpower,write mathematical models for inverse variation,write mathematical models for joint variation.

J. Robert Buchanan Mathematical Modeling and Variation

Direct Variation

The general linear model is represented by the equationy = mx + b.

If b = 0 so that y = mx we say that y varies directly with (or isdirectly proportional to) x .

Direct VariationThe following statements are equivalent.

1 y varies directly with x .2 y is directly proportional to x .3 y = k x for some nonzero constant k .

The constant k is called the constant of proportionality or theconstant of variation.

J. Robert Buchanan Mathematical Modeling and Variation

Direct Variation

The general linear model is represented by the equationy = mx + b.

If b = 0 so that y = mx we say that y varies directly with (or isdirectly proportional to) x .

Direct VariationThe following statements are equivalent.

1 y varies directly with x .2 y is directly proportional to x .3 y = k x for some nonzero constant k .

The constant k is called the constant of proportionality or theconstant of variation.

J. Robert Buchanan Mathematical Modeling and Variation

Example

State sales tax is based on retail price. An item that sells for$189.99 has a sales tax of $11.40. Find a mathematical modelthat gives the amount of sales tax y in terms of the retail pricex . Use the model to find the sales tax on a $639.99 purchase.

Assuming that y = kx then

11.40 = k(189.99) ⇐⇒ k =11.40

189.99= 0.06.

The sales tax on the new purchase will be

y = (0.06)(639.99) = 38.40.

J. Robert Buchanan Mathematical Modeling and Variation

Example

State sales tax is based on retail price. An item that sells for$189.99 has a sales tax of $11.40. Find a mathematical modelthat gives the amount of sales tax y in terms of the retail pricex . Use the model to find the sales tax on a $639.99 purchase.Assuming that y = kx then

11.40 = k(189.99) ⇐⇒ k =11.40

189.99= 0.06.

The sales tax on the new purchase will be

y = (0.06)(639.99) = 38.40.

J. Robert Buchanan Mathematical Modeling and Variation

Example

State sales tax is based on retail price. An item that sells for$189.99 has a sales tax of $11.40. Find a mathematical modelthat gives the amount of sales tax y in terms of the retail pricex . Use the model to find the sales tax on a $639.99 purchase.Assuming that y = kx then

11.40 = k(189.99) ⇐⇒ k =11.40

189.99= 0.06.

The sales tax on the new purchase will be

y = (0.06)(639.99) = 38.40.

J. Robert Buchanan Mathematical Modeling and Variation

Direct Variation as an nth Power

Sometimes a variable changes directly with a power of anothervariable.

ExampleThe volume V of a cube varies directly with the 3rd power ofthe length of an edge of the cube.

Direct Variation as an nth PowerThe following statements are equivalent.

1 y varies directly as the nth power of x .2 y is directly proportional to the nth power of x .3 y = k xn for some constant k .

J. Robert Buchanan Mathematical Modeling and Variation

Direct Variation as an nth Power

Sometimes a variable changes directly with a power of anothervariable.

ExampleThe volume V of a cube varies directly with the 3rd power ofthe length of an edge of the cube.

Direct Variation as an nth PowerThe following statements are equivalent.

1 y varies directly as the nth power of x .2 y is directly proportional to the nth power of x .3 y = k xn for some constant k .

J. Robert Buchanan Mathematical Modeling and Variation

Example

Consider the solid particles which may be found in a river,stream, or creek. The diameter of the largest particle that thewater can move varies directly with the square of the velocity ofthe water.

1 Write a mathematical model relating the diameter of thelargest particle the water can move to the velocity of thewater.

d = kv2

where d is the diameter and v is the velocity.

2 By what factor does the diameter decrease if the velocity ofthe water is halved?

kv2 = d

k(v

2

)2= (kv2)

(14

)=

d4

J. Robert Buchanan Mathematical Modeling and Variation

Example

Consider the solid particles which may be found in a river,stream, or creek. The diameter of the largest particle that thewater can move varies directly with the square of the velocity ofthe water.

1 Write a mathematical model relating the diameter of thelargest particle the water can move to the velocity of thewater.

d = kv2

where d is the diameter and v is the velocity.2 By what factor does the diameter decrease if the velocity of

the water is halved?

kv2 = d

k(v

2

)2= (kv2)

(14

)=

d4

J. Robert Buchanan Mathematical Modeling and Variation

Example

Consider the solid particles which may be found in a river,stream, or creek. The diameter of the largest particle that thewater can move varies directly with the square of the velocity ofthe water.

1 Write a mathematical model relating the diameter of thelargest particle the water can move to the velocity of thewater.

d = kv2

where d is the diameter and v is the velocity.2 By what factor does the diameter decrease if the velocity of

the water is halved?

kv2 = d

k(v

2

)2= (kv2)

(14

)=

d4

J. Robert Buchanan Mathematical Modeling and Variation

Inverse Variation

Sometimes a variable increases while another decreases.

Inverse VariationThe following statements are equivalent.

1 y varies inversely as x .2 y is inversely proportional to x .

3 y =kx

for some constant k .

In some situations y =kxn and we say that y varies inversely

with the nth power of x .

J. Robert Buchanan Mathematical Modeling and Variation

Inverse Variation

Sometimes a variable increases while another decreases.

Inverse VariationThe following statements are equivalent.

1 y varies inversely as x .2 y is inversely proportional to x .

3 y =kx

for some constant k .

In some situations y =kxn and we say that y varies inversely

with the nth power of x .

J. Robert Buchanan Mathematical Modeling and Variation

Inverse Variation

Sometimes a variable increases while another decreases.

Inverse VariationThe following statements are equivalent.

1 y varies inversely as x .2 y is inversely proportional to x .

3 y =kx

for some constant k .

In some situations y =kxn and we say that y varies inversely

with the nth power of x .

J. Robert Buchanan Mathematical Modeling and Variation

Example

The frequency of vibration of a piano string varies directly asthe square root of the tension on the string and inversely as thelength of the string. The middle A string has a frequency of 440vibrations per second. Find the frequency of a string that has1.25 times as much tension and is 1.2 times as long.

A suitable mathematical model is

f = k√

TL

,

f is frequency, T is tension, and L is length. For the middle Astring

440 = k√

TA

LA⇐⇒ k =

440LA√TA

.

For the new string

f = k√

1.25TA

1.2LA=

440LA√TA

√1.25TA

1.2LA=

440√

1.251.2

= 409.9

vibrations per second.

J. Robert Buchanan Mathematical Modeling and Variation

Example

The frequency of vibration of a piano string varies directly asthe square root of the tension on the string and inversely as thelength of the string. The middle A string has a frequency of 440vibrations per second. Find the frequency of a string that has1.25 times as much tension and is 1.2 times as long.

A suitable mathematical model is

f = k√

TL

,

f is frequency, T is tension, and L is length. For the middle Astring

440 = k√

TA

LA⇐⇒ k =

440LA√TA

.

For the new string

f = k√

1.25TA

1.2LA=

440LA√TA

√1.25TA

1.2LA=

440√

1.251.2

= 409.9

vibrations per second.

J. Robert Buchanan Mathematical Modeling and Variation

Example

The frequency of vibration of a piano string varies directly asthe square root of the tension on the string and inversely as thelength of the string. The middle A string has a frequency of 440vibrations per second. Find the frequency of a string that has1.25 times as much tension and is 1.2 times as long.

A suitable mathematical model is

f = k√

TL

,

f is frequency, T is tension, and L is length. For the middle Astring

440 = k√

TA

LA⇐⇒ k =

440LA√TA

.

For the new string

f = k√

1.25TA

1.2LA=

440LA√TA

√1.25TA

1.2LA=

440√

1.251.2

= 409.9

vibrations per second.J. Robert Buchanan Mathematical Modeling and Variation

Joint Variation

Sometimes a variable changes directly with two (or more) othervariables.

Joint VariationThe following statements are equivalent.

1 z varies jointly as x and y .2 z is jointly proportional to x and y .3 z = k x y for some constant k .

In some situations z = k xn ym and we say that z varies jointlywith the nth power of x and the mth power of y .

J. Robert Buchanan Mathematical Modeling and Variation

Joint Variation

Sometimes a variable changes directly with two (or more) othervariables.

Joint VariationThe following statements are equivalent.

1 z varies jointly as x and y .2 z is jointly proportional to x and y .3 z = k x y for some constant k .

In some situations z = k xn ym and we say that z varies jointlywith the nth power of x and the mth power of y .

J. Robert Buchanan Mathematical Modeling and Variation

Joint Variation

Sometimes a variable changes directly with two (or more) othervariables.

Joint VariationThe following statements are equivalent.

1 z varies jointly as x and y .2 z is jointly proportional to x and y .3 z = k x y for some constant k .

In some situations z = k xn ym and we say that z varies jointlywith the nth power of x and the mth power of y .

J. Robert Buchanan Mathematical Modeling and Variation

Example

The maximum load that can be safely supported by a horizontalbeam varies jointly as the width of the beam and the square ofits depth, and inversely as the length of the beam.

1 Write down a mathematical model for this situation.

M =k W D2

Lwhere M is the maximum safe load and W , D, L arerespectively width, depth, and length for the beam.

2 Determine the changes in the maximum safe load whenwidth and depth of the beam are doubled.

k W D2

L= M

k (2W ) (2D)2

L= 8

(k W D2

L

)= 8M

The maximum safe load is increased by a factor of 8.

J. Robert Buchanan Mathematical Modeling and Variation

Example

The maximum load that can be safely supported by a horizontalbeam varies jointly as the width of the beam and the square ofits depth, and inversely as the length of the beam.

1 Write down a mathematical model for this situation.

M =k W D2

Lwhere M is the maximum safe load and W , D, L arerespectively width, depth, and length for the beam.

2 Determine the changes in the maximum safe load whenwidth and depth of the beam are doubled.

k W D2

L= M

k (2W ) (2D)2

L= 8

(k W D2

L

)= 8M

The maximum safe load is increased by a factor of 8.

J. Robert Buchanan Mathematical Modeling and Variation

Example

The maximum load that can be safely supported by a horizontalbeam varies jointly as the width of the beam and the square ofits depth, and inversely as the length of the beam.

1 Write down a mathematical model for this situation.

M =k W D2

Lwhere M is the maximum safe load and W , D, L arerespectively width, depth, and length for the beam.

2 Determine the changes in the maximum safe load whenwidth and depth of the beam are doubled.

k W D2

L= M

k (2W ) (2D)2

L= 8

(k W D2

L

)= 8M

The maximum safe load is increased by a factor of 8.J. Robert Buchanan Mathematical Modeling and Variation

Homework

Read Section 1.10.Exercises: 35, 39, 43, 47, . . . , 71, 75

J. Robert Buchanan Mathematical Modeling and Variation