Announcements. 3.3 Variation More work on the edge between English and Math… of course this means language…speaking fluent math/English (manglish –

Announcements

3.3 Variation

More work on the edge between English and Math…

of course this means language…speaking fluent math/English (manglish – which is like spanglish or franglais) requires some practice.

The word “function” has become mainstream English –

“a child’s behavior is a function of his age and his experience”…the behavior is the outcome

(second coordinate) and the age and experience are the inputs (first coordinates)

…we’ve restricted ourselves to a single input in the class but it need not be so in other applications.

Now we’ll look at another concept that has gone mainstream…variation…if you have a stream of outputs – the stock market values or someone’s weight over the days of a diet you may indicate the dependence of this output on one or many inputs in several ways.

You may simply assert dependence as above or you may indicate a relationship that is

“direct” or “inverse” (also: indirect)

Direct relationships mean that

the output goes up as the input goes up

Direct variation

Weight varies directly as caloric intake. (Also phrased “is proportional to”.)

 W = kCwhere k is a positive real number

 where W is weight and C is caloric intake

Inverse relationships mean that output goes down as input goes up:

   …there’s a situation of diminishing

returns on investments

Inverse or indirect

Much past 90 minutes, a student’s ability to learn decreases with increasing time in manglish (ability varies inversely with time over 90 minutes)

 T > 90 minutes A is ability and T is time

in minutes above 90 

kA

t

It happens that sometimes there is dependence on more that one input and we call that “varies jointly”.

 If income is jointly proportional to

hourly wage and hours worked…you know it’s a direct relationship and we’ll say

 I = k (W)(T) I is income,

W is hourly wage, and T is hours worked

Note that I keep putting in a “k”… this is called the constant of proportionality and it always appears in variation problem. Indeed at times, it is to be solved for. Let’s look at how:

P is directly proportional to C. If C = 5, then P = 2.5. What is the constant of proportionality?

Typical Problem

Popper 009, Q1

Another typical problem:

Q varies inversely with T. If T = 15, then Q = 1/3. What is the constant of proportionality?

Another:

M is jointly proportional with A and B. If A = 5 and B = 7, then M is 70. What is the constant of proportionality?

Popper 9, Q2

Now, on to some story problems:

The cost of tiling a room is directly proportional to its area. If a 60 square foot room costs $150 to tile, how much does an 80 square foot room cost?

 A. Write the formula using C for cost and

A for area B. Find the constant of proportionality, kC. Solve for the indicated quantity 

Popper 9, Q3

  The cost of buying a hard drive varies inversely with time in years since 1998. If a hard drive costs $90 now, what will it cost in 2 years?

 A. Write the formula using C for cost and T for years since 1998

  B. Find the constant of proportionality, k

  C. Solve for the indicated quantity

The cost of a custom-made stained glass window is directly proportional to its area. If a window measuring 2 feet by 3 feet is $300. How much is a window that is 5 feet by 7 feet?A. Write the formula…define your variables appropriatelyB. Solve for the constant of proportionality, k

  C. Solve for the indicated quantity

Popper 9, Q4

The cost of making a suit varies directly with the cost of the fabric and inversely with the number of tailors involved. If the fabric cost $360 and the number of tailors is 3, then the suit costs $600. If the fabric costs $450 and the number of tailors is 5, how much does the suit cost?

A. Write the formula…define your variables appropriately

B. Solve for the constant of proportionality

C. Solve for the indicated quantity

The cost of a rug from Manufacturer A is directly proportional to its size. If a 5 x 8 rug costs $140, how much does a 10 x 12 rug cost.

A. Write the formula…define your variables appropriately

B. Solve for the constant of proportionality

C. Solve for the indicated quantity

GRADER PROOFING

Define your variables clearly. Be consistent with them.Mark your answer clearly – write it as a sentence if need be.

Popper 9, Q5