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)

_{}, _{}