p.372 #1

 

 

Maximize area:

 

Constraint:

 

 

 

 

Increases for

Decreases for

maximum at

 

 

Total width: 150 ft

Length: 100 ft

 


p.372 #2

Minimize wall length:

 

Constraint:

 

 

 

 

Decreases for

Increases for

 minimum at

 

 

a) Rooms should be 14.289 ft by 24.495 ft

 

b) 10 rooms:

p.373 #3

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.


p.372 #4

 

Constraints:

 

Maximum circle size

 

 

 

 

 

 

Decreases for

Increases for

minimum at

 

b) maximum must occur at endpoint

maximum area when  (all fence used on circle)

p.372 #5

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


p.372 #7

Minimize cost:

 

Constraint:

 

 

Decreases for

Increases for

minimum at

 

 



p.372 #10

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

 

 


p.372 #11

 

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.

 

p.372 #13

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

 


p.372 #15

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


p.372 #17

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

p.372 #19

 

 

Find maximum by graphing A:

 

[0, 2] x [0, 1.5]

 

Maximum at (0.860, 1.122)

 

,