Problem: A machinist is required to manufacture a circular metal disk with area 1000 cm2.


What radius produces such a disk?
Consider the function for which the domain is the set of all possible radii for the disk and the
function values (outputs) are the corresponding areas of the circular metal disk (i.e., f(radius)=area).
Sketch below the graph of the function for a range of inputs near and including the ideal radius.
Clearly indicate the point on the graph having the ideal radius as the input.

Use your calculator to build a table of values near and including the ideal radius.
Radius values (from lower to higher)
Corresponding area values
Ideal radius

If the machinist is allowed an error tolerance of ±5 cm2 in the area of the disk, how close to the
ideal radius in part (a) must the machinist control the radius?

Complete the following table, indicating “does not exist” as necessary:
Allowed error tolerance in area of disk Allowable difference from ideal radius
±5 cm2
±1 cm2
±0.5 cm2
±0.1 cm2

Can the ‘allowable difference from ideal radius’ be found for any ‘allowed error tolerance in area of
disk’?
Problem: See the table below. Your 80-year-old grandmother makes the best homemade apple butter
ever, but pronounces that she has made her last batch. She gives you a generous share for your
enjoyment, but also gives you a small container in which to put some of your share to ship to your
cousin in California. She gives you $3 (she’s not ‘cheap’, she was born during the Great Depression) to
ship as much of it as possible to your cousin. You’re broke (college student) and besides that, want to
keep as much of the butter for yourself anyway (cousins don’t really count as family when it comes to
MeeMaw’s apple butter), but will honor her wishes and use the money she’s provided.

If the container and necessary packing/wrapping material amounts to exactly 3 oz in weight, how
much apple butter can you send to your cousin, given your postage allowance?

Consider the function for which the domain is the set of all possible package weights and the
function values (outputs) are the corresponding postage fees (i.e., f(weight)=cost). Sketch below the
graph of the function for a range of inputs near and including the ideal weight (remember, this
should include the container and packaging material). Clearly indicate the point on the graph having
the ideal weight as the input.

Provide a table of values near and including the ideal weight.
Package weights (from lower to higher)
Postal fees
Ideal weight

If you only spend the $3 allowed, how close to the ideal package weight do you have to be?

How is the function in this second (apple butter) application different from the function in the first
(circular disk) application? Revisit the graphs and tables for some insight.
First-Class Mail—Retail Single Piece
(Source: http://pe.usps.com/text/dmm300/Notice123.htm)
Weight
Not Over
(ounces)
Letters
and Cards1
Large
Envelopes2
Packages3
1
$0.44
$0.88
$1.71
2
0.64
1.08
1.71
3
0.84
1.28
1.71
3.54
1.04
--
--
4
--
1.48
1.88
5
--
1.68
2.05
6
--
1.88
2.22
7
--
2.08
2.39
8
--
2.28
2.56
9
--
2.48
2.73
10
--
2.68
2.90
11
--
2.88
3.07
12
--
3.08
3.24
13
--
3.28
3.41
Postcard5
$0.29
--
--
1. Letters that meet one or more of the nonmachinable characteristics in DMM 101.1.2 are subject to the $0.20 nonmachinable surcharge.
2. Large envelope-sized pieces that are rigid, nonrectangular, or not uniformly thick pay package prices.
3. For keys and identification devices, add $0.75. If more than 13 ounces, see Priority Mail (footnote #4) on page 3.
4. For weights over 3.5 ounces, see large envelope (flat) prices.
5. The card price applies to each single or double postcard-sized mailpiece when originally mailed; reply half of double postcard must be
designed for reply mail purposes only.
See "" for postcard, letter, large envelope, and package dimensions.