Sect. 1.2 (part 1).notebook
September 25, 2012
Finding Minimum and Maximum Values
W
L
50 m of border is available to fence the three sides of the pen. What dimensions give the greatest area?
**Find the equation of best fit that models the relationship between the width and the area.
Width:
Area:
5
200
6
228
7
252
8
272
9
288
10
300
Sep 13­7:39 AM
1
Sect. 1.2 (part 1).notebook
September 25, 2012
Finding Minimum and Maximum Values
W
L
50 m of border is available to fence the three sides of the pen. What dimensions give the greatest area?
**Find the equation of best fit that models the relationship between the width and the area.
Width:
Area:
5
200
6
228
7
252
8
272
9
288
10
300
Pull
Pull
Use the table feature or graph to determine the
greatest area.
The greatest area is 312.5 m2 and is achieved when the width is 12.5 m
May 5­3:45 PM
2
Sect. 1.2 (part 1).notebook
September 25, 2012
Alternate Method (algebraically)
W
L
Perimeter = 2W + L
50
= 2W + L
50 - 2W = L
Solve the equation for one of the variables and substitute it into the equation you are trying to maximize.
Area = length X width
A
= (50 -2w)w
A
= 50w-2w2
Area is expressed as a function of its width.
Graph: y = 50x ­ 2x2
Use window: xmin: ­10
xmax: 30
ymin: ­100
ymax: 400
2nd Trace (Calc)
4: Maximum (this parabola has a maximum value at its vertex)
Move left, move right, enter
Maximum:
x = 12.5
When at maximum
y = 312.5
Maximum Value
May 5­4:12 PM
3
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
Find two numbers whose difference is 100 and
whose product is a minimum.
May 5­4:27 PM
4
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
A rectangular lot is bounded on one side by a
river and on the other three sides by a total
of 80 m of fencing. Find the dimensions of the
largest possible lot.
May 5­4:39 PM
5
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
A rectangular area is enclosed by a fence and
divided by another section of fence parallel to
two of its sides. If the 600m of fence used
encloses a maximum area, what are the
dimensions of the enclosure?
May 5­4:41 PM
6
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
A rectangular area is enclosed by a fence and
divided by another section of fence parallel to
two of its sides. If the 600m of fence used
encloses a maximum area, what are the
dimensions of the enclosure?
Sep 14­11:17 AM
7
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
A Newfoundland Theatre by the Sea group sells
all 400 tickets each night if it charges $4 per
ticket. The manager estimates that the ticket
sales would decrease by 25 for each $1.00
increase in the ticket price. What price would
result in the greatest revenue for the group?
May 5­4:45 PM
8
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
An ice cream store sells 1200 cones per day when
the price is 90 cents. Market research indicates
that for every 5 cent increase in price there will
be 100 fewer sales. What price, to the nearest
nickel, will produce the highest revenue from sales
of ice cream cones?
May 5­4:48 PM
9
Sect. 1.2 (part 1).notebook
Example:
September 25, 2012
Two consecutive odd numbers and a third number
add up to 48. What are the numbers so that the
sum of their squares is a minimum?
May 5­4:51 PM
10
Sect. 1.2 (part 1).notebook
September 25, 2012
A day care centre bought 20 m of board to form two sides of a rectangular sandbox against a corner in a fenced yard as shown. If all 20 m of board is used, write the quadratic function that models the area of the sandbox, and use it to determine the maximum area the sandbox can have.
Sep 14­8:54 AM
11
Sect. 1.2 (part 1).notebook
September 25, 2012
Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ? There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees ? A lifeguard marks off a rectangular swimming area using 120m of rope. If the beach is used as one side, what are the dimensions that give the maximum swimming area?
Find two numbers that have a sum of 36 and whose product is a maximum.
Stefan buys 20 m of chicken wire. He uses this wire to enclose a rectangular area for his rabbits. What are the dimensions of the enclosure that will give the greatest possible area? What is the maximum area?
Sep 13­7:39 AM
12