Maximize area:
Constraint:
Increases for
Decreases for
maximum at
Total width: 150 ft
Length: 100 ft
Minimize wall length:
Constraint:
Decreases for
Increases for
minimum at
a) Rooms should be 14.289 ft by 24.495 ft
b) 10 rooms:
Maximize area:
Constraints:
a) smallest when so
largest when using all fence except minimum used by square (40 ft) so
b)
[0, 100] x [0, 20000]
c)
Decreases for
Increases for
minimum at
Looking for maximum, so consider endpoints:
maximum area is 17,522.222 sq. ft.
Constraints:
Maximum circle size
Decreases for
Increases for
minimum at
b) maximum must occur at endpoint
maximum area when (all fence used on circle)
Maximize volume:
Constraint:
Increases for
Decreases for
maximum at
a) 6.325 cm by 6.325 cm by 3.162 cm
b) conjecture: Depth is half of width
Minimize cost:
Constraint:
Decreases for
Increases for
minimum at
Minimize area:
Constraints:
Increasing for
Decreasing for
maximum at
Minimum must occur at endpoint
[0, 150] x [0, 15000]
Smallest r is 20.
Largest r occurs when least straight lengths are used (i.e. ).
minimum occurs at
Minimize length of ladder, l:
Relate x and y using similar triangles:
Re-write l in one variable:
Find minimum using derivative:
(minimum is equivalent to minimum l)
*YUCK! TIME TO USE MY CALCULATOR!*
Graph :
[0, 12] x [0, 200]
Minimum occurs at (5, 125)
Since minimum is 125, minimum l is approximately 11.180 feet.
Maximize volume of cylinder:
Relate r and h using given perimeter:
Re-write V in one variable:
Find maximum using derivative:
Max occurs when , so
400 mm radius and 200 mm height
a)
b)
Relate r and h:
Rewrite A in terms of r:
c) Find minimum using derivative:
4.133 cm radius, 8.266 cm height
Can is short and fat
Ratio of diameter to altitude is 1 to 1
d)
The normal can uses close to the same amount of metal
Normal can uses about 1.5% more metal
e)
about $6.4 million
a)
Relate r and h together:
Re-write A in terms of one variable:
Minimize A:
Dimensions: radius 3.524 cm, height 3.524 cm
b) Ratio of diameter to altitude is 2 to 1
c)
Savings per year: $754,299.93
d) left to you J
Find maximum by graphing A:
[0, 2] x [0, 1.5]
Maximum at (0.860, 1.122)
,